pdf2htmlEX/demo/cheat3.page

<div class="pd w0 h0"><div id="pf3" class="pf" data-page-no="3"><div class="pc pc3"><img class="bi x0 y0 w2 h1" alt="" 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"/><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">Theoretical<span class="_ _0"> </span>Computer<span class="_ _0"> </span>Science<span class="_ _0"> </span>Cheat<span class="_ _0"> </span>Sheet</div><div class="t m0 xc6 h4 y232 ff3 fs0 fc0 sc0 ls0 ws0">π <span class="ff4">≈<span class="_ _7"> </span><span class="ff2">3</span></span>.<span class="ff2">14159,<span class="_ _38"> </span></span>e<span class="_ _7"> </span><span class="ff4">≈<span class="_ _7"> </span><span class="ff2">2</span></span>.<span class="ff2">71828,<span class="_ _38"> </span></span>γ <span class="ff4">≈<span class="_ _7"> </span><span class="ff2">0</span></span>.<span class="ff2">57721,<span class="_ _39"> </span></span>φ<span class="_ _7"> </span><span class="ff2">=</span></div><div class="t m0 x15c h5 y233 ff5 fs1 fc0 sc0 ls0 ws0">1+</div><div class="t m0 x15a h5 y234 ff8 fs1 fc0 sc0 ls0 ws0">√</div><div class="t m0 xb5 h5 y233 ff5 fs1 fc0 sc0 ls0 ws0">5</div><div class="t m0 x75 h5 y235 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x2e h4 y232 ff4 fs0 fc0 sc0 ls0 ws0">≈<span class="_ _7"> </span><span class="ff2">1<span class="ff3">.</span>61803,</span></div><div class="t m0 xb6 h3 y236 ff2 fs0 fc0 sc0 ls0 ws0">ˆ</div><div class="t m0 x13b h3 y232 ff3 fs0 fc0 sc0 ls0 ws0">φ<span class="_ _7"> </span><span class="ff2">=</span></div><div class="t m0 xb0 h5 y233 ff5 fs1 fc0 sc0 ls0 ws0">1<span class="ff8">−</span></div><div class="t m0 x6f h5 y234 ff8 fs1 fc0 sc0 ls0 ws0">√</div><div class="t m0 x19 h5 y233 ff5 fs1 fc0 sc0 ls0 ws0">5</div><div class="t m0 x16a h5 y235 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x52 h4 y232 ff4 fs0 fc0 sc0 ls1 ws0">≈−<span class="_ _b"></span><span class="ff3 ls0">.<span class="ff2">61803</span></span></div><div class="t m0 x16c h3 y237 ff3 fs0 fc0 sc0 ls0 ws0">i<span class="_ _3a"> </span><span class="ff2">2</span></div><div class="t m0 x107 h5 y238 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x14b h3 y237 ff3 fs0 fc0 sc0 ls0 ws0">p</div><div class="t m0 xfd h5 y239 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x25 h3 y237 ff2 fs0 fc0 sc0 ls0 ws0">General<span class="_ _3b"> </span>Probabilit<span class="_ _5"></span>y</div><div class="t m0 x81 h3 y23a ff2 fs0 fc0 sc0 ls0 ws0">1<span class="_ _3c"> </span>2<span class="_ _1d"> </span>2</div><div class="t m0 x13d h3 y11d ff2 fs0 fc0 sc0 ls0 ws0">Bernoulli Num<span class="_ _5"></span>b<span class="_ _3"></span>ers (<span class="ff3">B</span></div><div class="t m0 xef h5 y23b ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x137 h4 y11d ff2 fs0 fc0 sc0 ls1 ws0">=0<span class="_ _b"></span>,<span class="_ _15"></span>o<span class="_ _9"></span>d<span class="_ _b"></span>d<span class="_ _3"></span><span class="ff3 ls0">i<span class="_ _7"> </span><span class="ff4"><span class="ff2">=<span class="_ _7"> </span>1):</span></span></span></div><div class="t m0 xc3 h3 y23c ff3 fs0 fc0 sc0 ls0 ws0">B</div><div class="t m0 x16d h5 y23d ff5 fs1 fc0 sc0 ls0 ws0">0</div><div class="t m0 xca h3 y23e ff2 fs0 fc0 sc0 ls1 ws0">=1<span class="_ _b"></span>,<span class="_ _15"></span><span class="ff3 ls0">B</span></div><div class="t m0 x157 h5 y23d ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x11b h4 y23e ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff4">−</span></div><div class="t m0 xa h5 y23f ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 xa h5 y240 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x15e h3 y23e ff2 fs0 fc0 sc0 ls0 ws0">, <span class="ff3">B</span></div><div class="t m0 xef h5 y23d ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x73 h3 y23e ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x47 h5 y23f ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x47 h5 y240 ff5 fs1 fc0 sc0 ls0 ws0">6</div><div class="t m0 x14e h3 y23e ff2 fs0 fc0 sc0 ls0 ws0">, <span class="ff3">B</span></div><div class="t m0 x48 h5 y23d ff5 fs1 fc0 sc0 ls0 ws0">4</div><div class="t m0 x2c h4 y23e ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff4">−</span></div><div class="t m0 x15a h5 y23f ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x75 h5 y240 ff5 fs1 fc0 sc0 ls0 ws0">30</div><div class="t m0 xb5 h3 y23e ff2 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 xfe h3 y15 ff3 fs0 fc0 sc0 ls0 ws0">B</div><div class="t m0 xf8 h5 y12a ff5 fs1 fc0 sc0 ls0 ws0">6</div><div class="t m0 x8 h3 y15 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 xf9 h5 y19 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 xdc h5 y241 ff5 fs1 fc0 sc0 ls0 ws0">42</div><div class="t m0 x16e h3 y15 ff2 fs0 fc0 sc0 ls0 ws0">, <span class="ff3">B</span></div><div class="t m0 x25 h5 y12a ff5 fs1 fc0 sc0 ls0 ws0">8</div><div class="t m0 xa h4 y15 ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff4">−</span></div><div class="t m0 x72 h5 y19 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x16f h5 y241 ff5 fs1 fc0 sc0 ls0 ws0">30</div><div class="t m0 xc1 h3 y15 ff2 fs0 fc0 sc0 ls0 ws0">, <span class="ff3">B</span></div><div class="t m0 x49 h5 y12a ff5 fs1 fc0 sc0 ls0 ws0">10</div><div class="t m0 x68 h3 y15 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 xab h5 y19 ff5 fs1 fc0 sc0 ls0 ws0">5</div><div class="t m0 x3b h5 y241 ff5 fs1 fc0 sc0 ls0 ws0">66</div><div class="t m0 xac h3 y15 ff2 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x13d h3 y242 ff2 fs0 fc0 sc0 ls0 ws0">Change of base, quadratic formula:</div><div class="t m0 xc3 h3 y55 ff2 fs0 fc0 sc0 ls0 ws0">log</div><div class="t m0 xf6 h5 y243 ff6 fs1 fc0 sc0 ls0 ws0">b</div><div class="t m0 xc4 h3 y55 ff3 fs0 fc0 sc0 ls0 ws0">x<span class="_ _7"> </span><span class="ff2">=</span></div><div class="t m0 x15d h3 y244 ff2 fs0 fc0 sc0 ls0 ws0">log</div><div class="t m0 x11b h5 y14f ff6 fs1 fc0 sc0 ls0 ws0">a</div><div class="t m0 xfa h3 y245 ff3 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 xdc h3 y246 ff2 fs0 fc0 sc0 ls0 ws0">log</div><div class="t m0 x151 h5 y247 ff6 fs1 fc0 sc0 ls0 ws0">a</div><div class="t m0 x153 h3 y246 ff3 fs0 fc0 sc0 ls0 ws0">b</div><div class="t m0 x26 h3 y55 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x72 h4 y248 ff4 fs0 fc0 sc0 ls0 ws0">−<span class="ff3">b<span class="_ _8"> </span></span>±</div><div class="t m0 x5b h4 y249 ff4 fs0 fc0 sc0 ls0 ws0">√</div><div class="t m0 x126 h3 y245 ff3 fs0 fc0 sc0 ls0 ws0">b</div><div class="t m0 x48 h5 y24a ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x2c h4 y245 ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff2">4<span class="ff3">ac</span></span></div><div class="t m0 x156 h3 y246 ff2 fs0 fc0 sc0 ls0 ws0">2<span class="ff3">a</span></div><div class="t m0 x3d h3 y55 ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x13d h3 y59 ff2 fs0 fc0 sc0 ls0 ws0">Euler’s n<span class="_ _5"></span>umber <span class="ff3">e</span>:</div><div class="t m0 xca h3 y24b ff3 fs0 fc0 sc0 ls0 ws0">e<span class="_ _7"> </span><span class="ff2 ls1">=1<span class="_ _5"></span>+</span></div><div class="t m0 x11b h5 y24c ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x11b h5 y24d ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 xfa h3 y24e ff2 fs0 fc0 sc0 ls0 ws0">+</div><div class="t m0 x154 h5 y24c ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x154 h5 y24d ff5 fs1 fc0 sc0 ls0 ws0">6</div><div class="t m0 x9b h3 y24e ff2 fs0 fc0 sc0 ls0 ws0">+</div><div class="t m0 x16f h5 y24c ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 xef h5 y24d ff5 fs1 fc0 sc0 ls0 ws0">24</div><div class="t m0 xc1 h3 y24e ff2 fs0 fc0 sc0 ls0 ws0">+</div><div class="t m0 x5b h5 y24c ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x159 h5 y24d ff5 fs1 fc0 sc0 ls0 ws0">120</div><div class="t m0 x48 h4 y24e ff2 fs0 fc0 sc0 ls0 ws0">+<span class="_ _8"> </span><span class="ff4 ls4">···</span></div><div class="t m0 x15d h3 y24f ff2 fs0 fc0 sc0 ls0 ws0">lim</div><div class="t m0 xe8 h5 y250 ff6 fs1 fc0 sc0 ls0 ws0">n<span class="ff8">→∞</span></div><div class="t m0 xfa h6 y251 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x57 h3 y24f ff2 fs0 fc0 sc0 ls2 ws0">1+</div><div class="t m0 x45 h3 y252 ff3 fs0 fc0 sc0 ls0 ws0">x</div><div class="t m0 x45 h3 y5e ff3 fs0 fc0 sc0 ls0 ws0">n</div><div class="t m0 xef h6 y253 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x29 h5 y254 ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 xd h3 y255 ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff3">e</span></div><div class="t m0 x156 h5 y256 ff6 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 x68 h3 y255 ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 xc4 h6 y142 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xcb h3 y257 ff2 fs0 fc0 sc0 ls2 ws0">1+</div><div class="t m0 x157 h5 y258 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 xf9 h5 y32 ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 xdb h6 y142 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x151 h5 y30 ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x155 h3 y259 ff3 fs0 fc0 sc0 ls1 ws0">&lt;e&lt;</div><div class="t m0 xef h6 y142 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x137 h3 y259 ff2 fs0 fc0 sc0 ls2 ws0">1+</div><div class="t m0 x49 h5 y258 ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 x49 h5 y32 ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x156 h6 y142 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x126 h5 y30 ff6 fs1 fc0 sc0 ls0 ws0">n<span class="ff5">+1</span></div><div class="t m0 x5f h3 y259 ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x1 h6 y25a ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xc3 h3 y159 ff2 fs0 fc0 sc0 ls2 ws0">1+</div><div class="t m0 xc4 h5 y25b ff5 fs1 fc0 sc0 ls0 ws0">1</div><div class="t m0 xc4 h5 y25c ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x7 h6 y25d ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x8 h5 y25e ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x150 h4 y159 ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff3">e<span class="_ _8"> </span><span class="ff4">−</span></span></div><div class="t m0 x154 h3 y31 ff3 fs0 fc0 sc0 ls0 ws0">e</div><div class="t m0 x26 h3 y25f ff2 fs0 fc0 sc0 ls0 ws0">2<span class="ff3">n</span></div><div class="t m0 x12e h3 y159 ff2 fs0 fc0 sc0 ls0 ws0">+</div><div class="t m0 x137 h3 y31 ff2 fs0 fc0 sc0 ls0 ws0">11<span class="ff3">e</span></div><div class="t m0 x72 h3 y25f ff2 fs0 fc0 sc0 ls0 ws0">24<span class="ff3">n</span></div><div class="t m0 x2b h5 y25c ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x5b h4 y159 ff4 fs0 fc0 sc0 ls0 ws0">−<span class="_ _8"> </span><span class="ff3">O</span></div><div class="t m0 x2c h6 y260 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x15c h3 y31 ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 x170 h3 y25f ff3 fs0 fc0 sc0 ls0 ws0">n</div><div class="t m0 x12b h5 y25c ff5 fs1 fc0 sc0 ls0 ws0">3</div><div class="t m0 x75 h6 y260 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xb5 h3 y159 ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x13d h3 y261 ff2 fs0 fc0 sc0 ls0 ws0">Harmonic n<span class="_ _5"></span>umbers:</div><div class="t m0 x5 h3 y66 ff2 fs0 fc0 sc0 ls0 ws0">1,</div><div class="t m0 x8e h5 y262 ff5 fs1 fc0 sc0 ls0 ws0">3</div><div class="t m0 x8e h5 y263 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x171 h3 y66 ff2 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 xf8 h5 y262 ff5 fs1 fc0 sc0 ls0 ws0">11</div><div class="t m0 xcb h5 y263 ff5 fs1 fc0 sc0 ls0 ws0">6</div><div class="t m0 xda h3 y66 ff2 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x15d h5 y262 ff5 fs1 fc0 sc0 ls0 ws0">25</div><div class="t m0 x15d h5 y263 ff5 fs1 fc0 sc0 ls0 ws0">12</div><div class="t m0 x172 h3 y66 ff2 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 xeb h5 y262 ff5 fs1 fc0 sc0 ls0 ws0">137</div><div class="t m0 x173 h5 y263 ff5 fs1 fc0 sc0 ls0 ws0">60</div><div class="t m0 x44 h3 y66 ff2 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x9 h5 y262 ff5 fs1 fc0 sc0 ls0 ws0">49</div><div class="t m0 x9 h5 y263 ff5 fs1 fc0 sc0 ls0 ws0">20</div><div class="t m0 x9c h3 y66 ff2 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x72 h5 y262 ff5 fs1 fc0 sc0 ls0 ws0">363</div><div class="t m0 x72 h5 y263 ff5 fs1 fc0 sc0 ls0 ws0">140</div><div class="t m0 xd h3 y66 ff2 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x49 h5 y262 ff5 fs1 fc0 sc0 ls0 ws0">761</div><div class="t m0 x49 h5 y263 ff5 fs1 fc0 sc0 ls0 ws0">280</div><div class="t m0 x68 h3 y66 ff2 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x5c h5 y262 ff5 fs1 fc0 sc0 ls0 ws0">7129</div><div class="t m0 x5c h5 y263 ff5 fs1 fc0 sc0 ls0 ws0">2520</div><div class="t m0 x12b h3 y66 ff3 fs0 fc0 sc0 ls4 ws0">,...</div><div class="t m0 xe8 h3 y264 ff2 fs0 fc0 sc0 ls0 ws0">ln<span class="_ _6"> </span><span class="ff3 ls1">n&lt;H</span></div><div class="t m0 xea h5 y265 ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x12e h3 y266 ff3 fs0 fc0 sc0 ls0 ws0">&lt;<span class="_ _7"> </span><span class="ff2">ln<span class="_ _6"> </span></span>n<span class="_ _8"> </span><span class="ff2 ls2">+1<span class="_ _9"></span><span class="ff3 ls0">,</span></span></div><div class="t m0 x136 h3 y267 ff3 fs0 fc0 sc0 ls0 ws0">H</div><div class="t m0 xe8 h5 y1f4 ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x174 h3 y268 ff2 fs0 fc0 sc0 ls1 ws0">=l<span class="_ _b"></span>n<span class="_ _3d"></span><span class="ff3 ls0">n<span class="_ _8"> </span><span class="ff2">+<span class="_ _8"> </span></span>γ<span class="_ _7"> </span><span class="ff2">+<span class="_ _8"> </span></span>O</span></div><div class="t m0 x47 h6 y269 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x2a h3 y26a ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 x2a h3 y6e ff3 fs0 fc0 sc0 ls0 ws0">n</div><div class="t m0 x68 h6 y269 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x175 h3 y267 ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x13d h3 y26b ff2 fs0 fc0 sc0 ls0 ws0">F<span class="_ _5"></span>actorial,<span class="_ _7"> </span>Stirling’s approximation:</div><div class="t m0 x176 h3 y16a ff5 fs1 fc0 sc0 ls0 ws0">1,<span class="_ _7"> </span>2,<span class="_ _7"> </span>6,<span class="_ _7"> </span>24,<span class="_ _8"> </span>120,<span class="_ _7"> </span>720,<span class="_ _7"> </span>5040,<span class="_ _7"> </span>40320,<span class="_ _7"> </span>362880,<span class="_ _7"> </span><span class="ff3 fs0 ls4">...</span></div><div class="t m0 xf7 h3 y1bf ff3 fs0 fc0 sc0 ls0 ws0">n<span class="ff2 ls1">!=</span></div><div class="t m0 x14f h4 yab ff4 fs0 fc0 sc0 ls0 ws0">√</div><div class="t m0 xf9 h3 y1bf ff2 fs0 fc0 sc0 ls0 ws0">2<span class="ff3 ls10">πn</span></div><div class="t m0 xfa h6 y26c ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x27 h3 y1b6 ff3 fs0 fc0 sc0 ls0 ws0">n</div><div class="t m0 x44 h3 y16e ff3 fs0 fc0 sc0 ls0 ws0">e</div><div class="t m0 x177 h6 y26c ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x12e h5 y26d ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 xc h6 y26c ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x137 h3 y1bf ff2 fs0 fc0 sc0 ls2 ws0">1+Θ</div><div class="t m0 x156 h6 y26c ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x178 h3 y1b6 ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 x48 h3 y16e ff3 fs0 fc0 sc0 ls0 ws0">n</div><div class="t m0 x2c h6 y26c ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x12b h3 y1bf ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x13d h3 y26e ff2 fs0 fc0 sc0 ls0 ws0">Ac<span class="_ _5"></span>kermann’s function and in<span class="_ _5"></span>v<span class="_ _5"></span>erse:</div><div class="t m0 x1 h3 y1c5 ff3 fs0 fc0 sc0 ls0 ws0">a<span class="ff2">(</span>i,<span class="_ _6"> </span>j<span class="_ _15"></span><span class="ff2 lsf">)=</span></div><div class="t m0 x179 h6 y8f ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x179 h6 y8d ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x179 h6 y171 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x174 h3 y26f ff2 fs0 fc0 sc0 ls0 ws0">2</div><div class="t m0 x16e h5 y270 ff6 fs1 fc0 sc0 ls0 ws0">j</div><div class="t m0 x3b h3 y271 ff3 fs0 fc0 sc0 ls0 ws0">i<span class="_ _7"> </span><span class="ff2 ls1">=1</span></div><div class="t m0 x174 h4 y1c5 ff3 fs0 fc0 sc0 ls0 ws0">a<span class="ff2">(</span>i<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span><span class="ff2">1</span></span>,<span class="_ _6"> </span><span class="ff2">2)<span class="_ _3e"> </span></span>j <span class="ff2 ls1">=1</span></div><div class="t m0 x174 h4 y1c4 ff3 fs0 fc0 sc0 ls0 ws0">a<span class="ff2">(</span>i<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span><span class="ff2">1</span></span><span class="ls4">,a<span class="_ _1f"></span><span class="ff2 ls0">(<span class="_ _5"></span><span class="ff3">i,<span class="_ _8"> </span>j<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span><span class="ff2">1))<span class="_ _28"> </span></span></span>i,<span class="_ _8"> </span>j<span class="_ _7"> </span><span class="ff4">≥<span class="_ _7"> </span><span class="ff2">2</span></span></span></span></span></div><div class="t m0 x176 h4 y272 ff3 fs0 fc0 sc0 ls0 ws0">α<span class="ff2">(</span>i<span class="ff2">)<span class="_ _7"> </span>=<span class="_ _7"> </span>min<span class="ff4">{</span></span>j <span class="ff4">|<span class="_ _7"> </span></span>a<span class="ff2">(</span><span class="ls11">j,<span class="_ _6"> </span>j<span class="_ _15"></span></span><span class="ff2">)<span class="_ _7"> </span><span class="ff4">≥<span class="_ _7"> </span></span></span>i<span class="ff4">}</span>.</div><div class="t m0 x3f h3 y273 ff2 fs0 fc0 sc0 ls0 ws0">Con<span class="_ _5"></span>tinuous distributions:<span class="_ _34"> </span>If</div><div class="t m0 xa0 h3 y274 ff2 fs0 fc0 sc0 ls0 ws0">Pr[<span class="ff3 ls1">a&lt;X<span class="_ _15"></span>&lt;b<span class="_ _b"></span><span class="ff2">]=</span></span></div><div class="t m0 xbb h6 y4f ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xcf h5 y275 ff6 fs1 fc0 sc0 ls0 ws0">b</div><div class="t m0 x64 h5 y15 ff6 fs1 fc0 sc0 ls0 ws0">a</div><div class="t m0 xbc h3 y276 ff3 fs0 fc0 sc0 ls0 ws0">p<span class="ff2">(</span>x<span class="ff2">)<span class="_ _6"> </span></span>dx,</div><div class="t m0 x3f h3 y277 ff2 fs0 fc0 sc0 ls0 ws0">then<span class="_ _7"> </span><span class="ff3">p<span class="_ _7"> </span></span>is<span class="_ _7"> </span>the<span class="_ _14"> </span>probabilit<span class="_ _5"></span>y<span class="_ _7"> </span>density<span class="_ _7"> </span>function<span class="_ _7"> </span>of</div><div class="t m0 x3f h3 y278 ff3 fs0 fc0 sc0 ls0 ws0">X<span class="_ _15"></span><span class="ff2 ls12">.I<span class="_ _3f"></span>f</span></div><div class="t m0 x11d h3 y1e ff2 fs0 fc0 sc0 ls0 ws0">Pr[<span class="ff3 ls13">X&lt;<span class="_ _5"></span>a<span class="_ _3f"></span><span class="ff2 ls1">]=<span class="ff3 ls0">P<span class="_ _6"> </span><span class="ff2">(</span>a<span class="ff2">)</span>,</span></span></span></div><div class="t m0 x3f h3 y279 ff2 fs0 fc0 sc0 ls0 ws0">then <span class="ff3">P<span class="_ _34"> </span></span>is the distribution<span class="_ _7"> </span>function of <span class="ff3">X<span class="_ _15"></span></span><span class="ls12">.I<span class="_ _3f"></span>f</span></div><div class="t m0 x3f h3 y27a ff3 fs0 fc0 sc0 ls0 ws0">P<span class="_ _1e"> </span><span class="ff2">and </span>p <span class="ff2">both exist then</span></div><div class="t m0 x4d h3 y27b ff3 fs0 fc0 sc0 ls0 ws0">P<span class="_ _6"> </span><span class="ff2">(</span>a<span class="ff2 ls1">)=</span></div><div class="t m0 x6e h6 y5a ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x17a h5 y25 ff6 fs1 fc0 sc0 ls0 ws0">a</div><div class="t m0 xb8 h5 y27c ff8 fs1 fc0 sc0 ls0 ws0">−∞</div><div class="t m0 xbb h3 y27d ff3 fs0 fc0 sc0 ls0 ws0">p<span class="ff2">(</span>x<span class="ff2">)<span class="_ _6"> </span></span>dx.</div><div class="t m0 x3f h3 y157 ff2 fs0 fc0 sc0 ls0 ws0">Exp<span class="_ _3"></span>ectation:<span class="_ _1e"> </span>If<span class="_ _7"> </span><span class="ff3">X<span class="_ _34"> </span></span>is discrete</div><div class="t m0 x4c h3 y27e ff2 fs0 fc0 sc0 ls0 ws0">E</div><div class="t m0 x17b h3 y27f ff2 fs0 fc0 sc0 ls0 ws0">[<span class="ff3">g<span class="_ _3"></span></span>(<span class="ff3">X<span class="_ _2"></span></span>)]<span class="_ _7"> </span>=</div><div class="t m0 xa9 h6 y34 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x4f h5 y280 ff6 fs1 fc0 sc0 ls0 ws0">x</div><div class="t m0 x1a h3 y281 ff3 fs0 fc0 sc0 ls0 ws0">g<span class="_ _3"></span><span class="ff2">(</span>x<span class="ff2 ls4">)P<span class="_ _1f"></span>r<span class="_ _1f"></span>[<span class="_ _9"></span><span class="ff3 ls0">X<span class="_ _0"> </span><span class="ff2">=<span class="_ _7"> </span></span>x<span class="ff2">]</span>.</span></span></div><div class="t m0 x3f h3 y25f ff2 fs0 fc0 sc0 ls0 ws0">If <span class="ff3">X<span class="_ _34"> </span></span>con<span class="_ _5"></span>tin<span class="_ _5"></span>uous then</div><div class="t m0 x3f h3 y282 ff2 fs0 fc0 sc0 ls0 ws0">E</div><div class="t m0 x6c h3 y3a ff2 fs0 fc0 sc0 ls0 ws0">[<span class="ff3">g<span class="_ _3"></span></span>(<span class="ff3">X<span class="_ _2"></span></span>)]<span class="_ _7"> </span>=</div><div class="t m0 x43 h6 y283 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x79 h5 y284 ff8 fs1 fc0 sc0 ls0 ws0">∞</div><div class="t m0 x11d h5 y285 ff8 fs1 fc0 sc0 ls0 ws0">−∞</div><div class="t m0 x7b h3 y3a ff3 fs0 fc0 sc0 ls0 ws0">g<span class="_ _3"></span><span class="ff2">(</span>x<span class="ff2">)</span>p<span class="ff2">(</span>x<span class="ff2">)<span class="_ _8"> </span></span>dx<span class="_ _8"> </span><span class="ff2">=</span></div><div class="t m0 x37 h6 y286 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x55 h5 y284 ff8 fs1 fc0 sc0 ls0 ws0">∞</div><div class="t m0 x7c h5 y285 ff8 fs1 fc0 sc0 ls0 ws0">−∞</div><div class="t m0 x7e h3 y3a ff3 fs0 fc0 sc0 ls0 ws0">g<span class="_ _3"></span><span class="ff2">(</span>x<span class="ff2">)<span class="_ _8"> </span></span>dP<span class="_ _2"></span><span class="ff2">(</span>x<span class="ff2">)</span>.</div><div class="t m0 x3f h3 y43 ff2 fs0 fc0 sc0 ls0 ws0">V<span class="_ _5"></span>ariance,<span class="_ _7"> </span>standard deviation:</div><div class="t m0 x17c h3 y69 ff2 fs0 fc0 sc0 ls0 ws0">V<span class="_ _3d"></span>AR[<span class="ff3">X<span class="_ _15"></span></span><span class="ls1">]=</span></div><div class="t m0 x113 h3 y287 ff2 fs0 fc0 sc0 ls0 ws0">E</div><div class="t m0 x1a h3 y69 ff2 fs0 fc0 sc0 ls0 ws0">[<span class="ff3">X</span></div><div class="t m0 x52 h5 y288 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x1c h4 y69 ff2 fs0 fc0 sc0 ls0 ws0">]<span class="_ _8"> </span><span class="ff4">−</span></div><div class="t m0 xd0 h3 y289 ff2 fs0 fc0 sc0 ls0 ws0">E</div><div class="t m0 x1f h3 y69 ff2 fs0 fc0 sc0 ls0 ws0">[<span class="ff3">X<span class="_ _15"></span></span>]</div><div class="t m0 x7e h5 y288 ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 xd1 h3 y69 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x7a h3 y6c ff3 fs0 fc0 sc0 ls0 ws0">σ <span class="ff2">=</span></div><div class="t m0 x113 h6 y269 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x36 h3 y6c ff2 fs0 fc0 sc0 ls14 ws0">VA<span class="_ _2"></span>R<span class="_ _2"></span>[<span class="_ _2"></span><span class="ff3 ls0">X<span class="_ _2"></span><span class="ff2">]</span>.</span></div><div class="t m0 x3f h3 y28a ff2 fs0 fc0 sc0 lsc ws0">F<span class="_ _5"></span>or<span class="_ _0"> </span>eve<span class="_ _3"></span>nt<span class="_ _3"></span>s<span class="_ _0"> </span><span class="ff3 ls0">A <span class="ff2">and </span>B<span class="_ _3"></span><span class="ff2">:</span></span></div><div class="t m0 x40 h4 y28b ff2 fs0 fc0 sc0 ls0 ws0">Pr[<span class="ff3">A<span class="_ _8"> </span><span class="ff4">∨<span class="_ _8"> </span></span>B<span class="_ _3"></span></span>]<span class="_ _7"> </span>=<span class="_ _7"> </span>Pr[<span class="ff3">A</span>]<span class="_ _8"> </span>+<span class="_ _8"> </span>Pr[<span class="ff3">B<span class="_ _15"></span></span>]<span class="_ _8"> </span><span class="ff4">−<span class="_ _8"> </span></span>Pr[<span class="ff3">A<span class="_ _8"> </span><span class="ff4">∧<span class="_ _8"> </span></span>B<span class="_ _3"></span></span>]</div><div class="t m0 x40 h4 y168 ff2 fs0 fc0 sc0 ls0 ws0">Pr[<span class="ff3">A<span class="_ _8"> </span><span class="ff4">∧<span class="_ _8"> </span></span>B<span class="_ _3"></span></span>]<span class="_ _7"> </span>=<span class="_ _7"> </span>Pr[<span class="ff3">A</span>]<span class="_ _8"> </span><span class="ff4">·<span class="_ _8"> </span></span>Pr[<span class="ff3">B<span class="_ _15"></span></span>]<span class="ff3">,</span></div><div class="t m0 x17d h3 y28c ff2 fs0 fc0 sc0 ls0 ws0">iff <span class="ff3">A </span>and <span class="ff3">B<span class="_ _0"> </span></span>are indep<span class="_ _3"></span>enden<span class="_ _5"></span>t.</div><div class="t m0 x77 h4 y28d ff2 fs0 fc0 sc0 ls0 ws0">Pr[<span class="ff3">A<span class="ff4">|</span>B<span class="_ _3"></span></span><span class="ls1">]=</span></div><div class="t m0 x7b h4 y28e ff2 fs0 fc0 sc0 ls0 ws0">Pr[<span class="ff3">A<span class="_ _8"> </span><span class="ff4">∧<span class="_ _8"> </span></span>B<span class="_ _3"></span></span>]</div><div class="t m0 x35 h3 y28f ff2 fs0 fc0 sc0 ls0 ws0">Pr[<span class="ff3">B<span class="_ _3"></span></span>]</div><div class="t m0 x3f h3 y290 ff2 fs0 fc0 sc0 ls0 ws0">F<span class="_ _5"></span>o<span class="_ _5"></span>r random v<span class="_ _5"></span>ariables<span class="_ _14"> </span><span class="ff3">X<span class="_ _34"> </span></span>and <span class="ff3">Y<span class="_ _8"> </span></span>:</div><div class="t m0 x13a h3 y291 ff2 fs0 fc0 sc0 ls0 ws0">E</div><div class="t m0 x4b h4 y93 ff2 fs0 fc0 sc0 ls0 ws0">[<span class="ff3">X<span class="_ _7"> </span><span class="ff4">·<span class="_ _8"> </span></span>Y<span class="_ _8"> </span></span><span class="ls1">]=</span></div><div class="t m0 x7b h3 y291 ff2 fs0 fc0 sc0 ls0 ws0">E</div><div class="t m0 x169 h4 y93 ff2 fs0 fc0 sc0 ls0 ws0">[<span class="ff3">X<span class="_ _15"></span></span>]<span class="_ _8"> </span><span class="ff4">·</span></div><div class="t m0 x36 h3 y291 ff2 fs0 fc0 sc0 ls0 ws0">E</div><div class="t m0 x63 h3 y93 ff2 fs0 fc0 sc0 ls0 ws0">[<span class="ff3">Y<span class="_ _8"> </span></span>]<span class="ff3">,</span></div><div class="t m0 x6d h3 y292 ff2 fs0 fc0 sc0 ls0 ws0">if <span class="ff3">X<span class="_ _34"> </span></span>and <span class="ff3">Y<span class="_ _11"> </span></span>are independent.</div><div class="t m0 xce h3 y293 ff2 fs0 fc0 sc0 ls0 ws0">E</div><div class="t m0 x122 h3 y177 ff2 fs0 fc0 sc0 ls0 ws0">[<span class="ff3">X<span class="_ _7"> </span></span>+<span class="_ _8"> </span><span class="ff3">Y<span class="_ _8"> </span></span><span class="ls1">]=</span></div><div class="t m0 x7b h3 y293 ff2 fs0 fc0 sc0 ls0 ws0">E</div><div class="t m0 x169 h3 y177 ff2 fs0 fc0 sc0 ls0 ws0">[<span class="ff3">X<span class="_ _15"></span></span><span class="ls2">]+</span></div><div class="t m0 x17e h3 y293 ff2 fs0 fc0 sc0 ls0 ws0">E</div><div class="t m0 xbb h3 y177 ff2 fs0 fc0 sc0 ls0 ws0">[<span class="ff3">Y<span class="_ _8"> </span></span>]<span class="ff3">,</span></div><div class="t m0 x32 h3 y294 ff2 fs0 fc0 sc0 ls0 ws0">E</div><div class="t m0 x111 h3 y295 ff2 fs0 fc0 sc0 ls0 ws0">[<span class="ff3">cX<span class="_ _15"></span></span><span class="ls1">]=</span><span class="ff3">c</span></div><div class="t m0 x61 h3 y294 ff2 fs0 fc0 sc0 ls0 ws0">E</div><div class="t m0 x50 h3 y295 ff2 fs0 fc0 sc0 ls0 ws0">[<span class="ff3">X<span class="_ _15"></span></span>]<span class="ff3">.</span></div><div class="t m0 x3f h3 y20b ff2 fs0 fc0 sc0 ls0 ws0">Ba<span class="_ _5"></span>yes’ theorem:</div><div class="t m0 x42 h3 yb8 ff2 fs0 fc0 sc0 ls0 ws0">Pr[<span class="ff3">A</span></div><div class="t m0 x162 h5 ybc ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x16b h4 yb8 ff4 fs0 fc0 sc0 ls0 ws0">|<span class="ff3">B<span class="_ _3"></span><span class="ff2 ls1">]=</span></span></div><div class="t m0 x113 h4 yba ff2 fs0 fc0 sc0 ls0 ws0">Pr[<span class="ff3">B<span class="_ _3"></span><span class="ff4">|</span>A</span></div><div class="t m0 xcf h5 y182 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x166 h3 yba ff2 fs0 fc0 sc0 ls4 ws0">]P<span class="_ _1f"></span>r<span class="_ _1f"></span>[<span class="_ _9"></span><span class="ff3 ls0">A</span></div><div class="t m0 xa4 h5 y182 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 xd1 h3 yba ff2 fs0 fc0 sc0 ls0 ws0">]</div><div class="t m0 x60 h6 y296 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x4f h5 y297 ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x4f h5 y298 ff6 fs1 fc0 sc0 ls0 ws0">j<span class="_ _3"></span><span class="ff5">=1</span></div><div class="t m0 x51 h3 ybb ff2 fs0 fc0 sc0 ls0 ws0">Pr[<span class="ff3">A</span></div><div class="t m0 x1e h5 y299 ff6 fs1 fc0 sc0 ls0 ws0">j</div><div class="t m0 xbc h4 ybb ff2 fs0 fc0 sc0 ls4 ws0">]P<span class="_ _1f"></span>r<span class="_ _1f"></span>[<span class="_ _9"></span><span class="ff3 ls0">B<span class="_ _15"></span><span class="ff4">|</span>A</span></div><div class="t m0 xbf h5 y299 ff6 fs1 fc0 sc0 ls0 ws0">j</div><div class="t m0 xba h3 ybb ff2 fs0 fc0 sc0 ls0 ws0">]</div><div class="t m0 x165 h3 yb8 ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x3f h3 y29a ff2 fs0 fc0 sc0 ls0 ws0">Inclusion-exclusion:</div><div class="t m0 x4a h3 y29b ff2 fs0 fc0 sc0 ls0 ws0">Pr</div><div class="t m0 x17f h6 y29c ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x4c h5 y188 ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x31 h6 y213 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x9f h5 y29d ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff5">=1</span></div><div class="t m0 x17 h3 y29e ff3 fs0 fc0 sc0 ls0 ws0">X</div><div class="t m0 x34 h5 y29f ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x4d h6 y2a0 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xec h3 y2a1 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x61 h5 y188 ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x7a h6 y213 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xa1 h5 y29d ff6 fs1 fc0 sc0 ls0 ws0">i<span class="ff5">=1</span></div><div class="t m0 x6e h3 y29e ff2 fs0 fc0 sc0 ls0 ws0">Pr[<span class="ff3">X</span></div><div class="t m0 x180 h5 y29f ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x1c h3 y29e ff2 fs0 fc0 sc0 ls2 ws0">]+</div><div class="t m0 x181 h5 y2a2 ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x162 h6 y2a3 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xdd h5 y2a4 ff6 fs1 fc0 sc0 ls0 ws0">k<span class="ff5">=2</span></div><div class="t m0 x11d h4 y2a5 ff2 fs0 fc0 sc0 ls0 ws0">(<span class="ff4">−</span>1)</div><div class="t m0 x35 h5 y2a6 ff6 fs1 fc0 sc0 ls0 ws0">k<span class="ff5">+1</span></div><div class="t m0 x63 h6 y2a3 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x6f h5 y195 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x17a h7 yd4 ffa fs2 fc0 sc0 ls0 ws0">i</div><div class="t m0 x19 h5 y195 ff6 fs1 fc0 sc0 ls0 ws0">&lt;<span class="ff8">···</span>&lt;i</div><div class="t m0 x1e h7 yd4 ffa fs2 fc0 sc0 ls0 ws0">k</div><div class="t m0 xa3 h3 y2a5 ff2 fs0 fc0 sc0 ls0 ws0">Pr</div><div class="t m0 x55 h6 y2a7 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xd1 h5 y2a2 ff6 fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 xa4 h6 y2a3 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x7e h5 y195 ff6 fs1 fc0 sc0 ls0 ws0">j<span class="_ _3"></span><span class="ff5">=1</span></div><div class="t m0 x182 h3 y2a5 ff3 fs0 fc0 sc0 ls0 ws0">X</div><div class="t m0 x183 h5 y2a8 ff6 fs1 fc0 sc0 ls0 ws0">i</div><div class="t m0 x184 h7 y2a9 ffa fs2 fc0 sc0 ls0 ws0">j</div><div class="t m0 x38 h6 y2aa ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x185 h3 y2ab ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x3f h3 y2ac ff2 fs0 fc0 sc0 ls0 ws0">Momen<span class="_ _5"></span>t inequalities:</div><div class="t m0 x16b h3 y1e3 ff2 fs0 fc0 sc0 ls0 ws0">Pr</div><div class="t m0 xec h6 y2ad ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xb6 h4 y1e3 ff4 fs0 fc0 sc0 ls0 ws0">|<span class="ff3">X<span class="_ _15"></span></span><span class="ls1">|≥</span><span class="ff3">λ</span></div><div class="t m0 x36 h3 y2ae ff2 fs0 fc0 sc0 ls0 ws0">E</div><div class="t m0 x70 h3 y1e3 ff2 fs0 fc0 sc0 ls0 ws0">[<span class="ff3">X<span class="_ _15"></span></span>]</div><div class="t m0 x54 h6 y2ad ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xd0 h4 y1e3 ff4 fs0 fc0 sc0 ls0 ws0">≤</div><div class="t m0 x20 h3 y2af ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 xf0 h3 y2b0 ff3 fs0 fc0 sc0 ls0 ws0">λ</div><div class="t m0 x65 h3 y1e3 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x110 h3 y2b1 ff2 fs0 fc0 sc0 ls0 ws0">Pr</div><div class="t m0 x162 h6 y2b2 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x186 h6 y2b3 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x186 h6 y2b4 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x43 h4 y2b1 ff3 fs0 fc0 sc0 ls0 ws0">X<span class="_ _7"> </span><span class="ff4">−</span></div><div class="t m0 x187 h3 y2b5 ff2 fs0 fc0 sc0 ls0 ws0">E</div><div class="t m0 xa9 h3 y2b1 ff2 fs0 fc0 sc0 ls0 ws0">[<span class="ff3">X<span class="_ _15"></span></span>]</div><div class="t m0 x6f h6 y2b6 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x6f h6 y2b7 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x19 h4 y2b8 ff4 fs0 fc0 sc0 ls0 ws0">≥<span class="_ _7"> </span><span class="ff3">λ<span class="_ _8"> </span></span>·<span class="_ _8"> </span><span class="ff3">σ</span></div><div class="t m0 x168 h6 y2b9 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xf0 h4 y2b8 ff4 fs0 fc0 sc0 ls0 ws0">≤</div><div class="t m0 xd1 h3 y1e4 ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 xa4 h3 y2ba ff3 fs0 fc0 sc0 ls0 ws0">λ</div><div class="t m0 xbe h5 y2bb ff5 fs1 fc0 sc0 ls0 ws0">2</div><div class="t m0 x167 h3 y2b8 ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x3f h3 y2bc ff2 fs0 fc0 sc0 ls0 ws0">Geometric distribution:</div><div class="t m0 x42 h3 y105 ff2 fs0 fc0 sc0 ls0 ws0">Pr[<span class="ff3">X </span>=<span class="_ _7"> </span><span class="ff3">k<span class="_ _15"></span></span><span class="ls1">]=</span><span class="ff3">pq</span></div><div class="t m0 x164 h5 yf8 ff6 fs1 fc0 sc0 ls0 ws0">k<span class="ff8">−<span class="ff5">1</span></span></div><div class="t m0 x63 h4 y105 ff3 fs0 fc0 sc0 lsd ws0">,q<span class="_ _40"></span><span class="ff2 ls1">=1<span class="_ _5"></span><span class="ff4 ls0">−<span class="_ _8"></span><span class="ff3">p,</span></span></span></div><div class="t m0 x33 h3 y2bd ff2 fs0 fc0 sc0 ls0 ws0">E</div><div class="t m0 x4d h3 y2be ff2 fs0 fc0 sc0 ls0 ws0">[<span class="ff3">X<span class="_ _15"></span></span><span class="ls1">]=</span></div><div class="t m0 x188 h5 y22b ff8 fs1 fc0 sc0 ls0 ws0">∞</div><div class="t m0 x18 h6 y2bf ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x18 h5 y2c0 ff6 fs1 fc0 sc0 ls0 ws0">k<span class="ff5">=1</span></div><div class="t m0 x1b h3 y2c1 ff3 fs0 fc0 sc0 ls10 ws0">kp<span class="_ _5"></span>q</div><div class="t m0 x1c h5 y109 ff6 fs1 fc0 sc0 ls0 ws0">k<span class="ff8">−<span class="ff5">1</span></span></div><div class="t m0 xa3 h3 y2c1 ff2 fs0 fc0 sc0 ls0 ws0">=</div><div class="t m0 x55 h3 y10c ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 x55 h3 y2c2 ff3 fs0 fc0 sc0 ls0 ws0">p</div><div class="t m0 x7e h3 y2c3 ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x81 h3 y2c4 ff2 fs0 fc0 sc0 ls0 ws0">2<span class="_ _3c"> </span>4<span class="_ _1d"> </span>3</div><div class="t m0 x81 h3 y2c5 ff2 fs0 fc0 sc0 ls0 ws0">3<span class="_ _3c"> </span>8<span class="_ _1d"> </span>5</div><div class="t m0 x81 h3 y13 ff2 fs0 fc0 sc0 ls0 ws0">4<span class="_ _41"> </span>16<span class="_ _42"> </span>7</div><div class="t m0 x81 h3 y132 ff2 fs0 fc0 sc0 ls0 ws0">5<span class="_ _41"> </span>32<span class="_ _43"> </span>11</div><div class="t m0 x81 h3 y20 ff2 fs0 fc0 sc0 ls0 ws0">6<span class="_ _41"> </span>64<span class="_ _43"> </span>13</div><div class="t m0 x81 h3 y2c6 ff2 fs0 fc0 sc0 ls0 ws0">7<span class="_ _3e"> </span>128<span class="_ _44"> </span>17</div><div class="t m0 x81 h3 y2c7 ff2 fs0 fc0 sc0 ls0 ws0">8<span class="_ _3e"> </span>256<span class="_ _44"> </span>19</div><div class="t m0 x81 h3 y2c8 ff2 fs0 fc0 sc0 ls0 ws0">9<span class="_ _3e"> </span>512<span class="_ _44"> </span>23</div><div class="t m0 x80 h3 y2c9 ff2 fs0 fc0 sc0 ls0 ws0">10<span class="_ _45"> </span>1,024<span class="_ _46"> </span>29</div><div class="t m0 x80 h3 y38 ff2 fs0 fc0 sc0 ls0 ws0">11<span class="_ _45"> </span>2,048<span class="_ _46"> </span>31</div><div class="t m0 x80 h3 y2ca ff2 fs0 fc0 sc0 ls0 ws0">12<span class="_ _45"> </span>4,096<span class="_ _46"> </span>37</div><div class="t m0 x80 h3 y15a ff2 fs0 fc0 sc0 ls0 ws0">13<span class="_ _45"> </span>8,192<span class="_ _46"> </span>41</div><div class="t m0 x80 h3 y2cb ff2 fs0 fc0 sc0 ls0 ws0">14<span class="_ _1a"> </span>16,384<span class="_ _47"> </span>43</div><div class="t m0 x80 h3 y2cc ff2 fs0 fc0 sc0 ls0 ws0">15<span class="_ _1a"> </span>32,768<span class="_ _47"> </span>47</div><div class="t m0 x80 h3 y47 ff2 fs0 fc0 sc0 ls0 ws0">16<span class="_ _1a"> </span>65,536<span class="_ _47"> </span>53</div><div class="t m0 x80 h3 y2cd ff2 fs0 fc0 sc0 ls0 ws0">17<span class="_ _33"> </span>131,072<span class="_ _13"> </span>59</div><div class="t m0 x80 h3 y2ce ff2 fs0 fc0 sc0 ls0 ws0">18<span class="_ _33"> </span>262,144<span class="_ _13"> </span>61</div><div class="t m0 x80 h3 y28b ff2 fs0 fc0 sc0 ls0 ws0">19<span class="_ _33"> </span>524,288<span class="_ _13"> </span>67</div><div class="t m0 x80 h3 y168 ff2 fs0 fc0 sc0 ls0 ws0">20<span class="_ _48"> </span>1,048,576<span class="_ _49"> </span>71</div><div class="t m0 x80 h3 y1fb ff2 fs0 fc0 sc0 ls0 ws0">21<span class="_ _48"> </span>2,097,152<span class="_ _49"> </span>73</div><div class="t m0 x80 h3 y28e ff2 fs0 fc0 sc0 ls0 ws0">22<span class="_ _48"> </span>4,194,304<span class="_ _49"> </span>79</div><div class="t m0 x80 h3 y2cf ff2 fs0 fc0 sc0 ls0 ws0">23<span class="_ _48"> </span>8,388,608<span class="_ _49"> </span>83</div><div class="t m0 x80 h3 y1c1 ff2 fs0 fc0 sc0 ls0 ws0">24<span class="_ _10"> </span>16,777,216<span class="_ _1a"> </span>89</div><div class="t m0 x80 h3 y90 ff2 fs0 fc0 sc0 ls0 ws0">25<span class="_ _10"> </span>33,554,432<span class="_ _1a"> </span>97</div><div class="t m0 x80 h3 y2d0 ff2 fs0 fc0 sc0 ls0 ws0">26<span class="_ _10"> </span>67,108,864<span class="_ _4a"> </span>101</div><div class="t m0 x80 h3 y2d1 ff2 fs0 fc0 sc0 ls0 ws0">27<span class="_ _4b"> </span>134,217,728<span class="_ _4c"> </span>103</div><div class="t m0 x80 h3 y9e ff2 fs0 fc0 sc0 ls0 ws0">28<span class="_ _4b"> </span>268,435,456<span class="_ _4c"> </span>107</div><div class="t m0 x13d h3 y9e ff2 fs0 fc0 sc0 ls0 ws0">Binomial distribution:</div><div class="t m0 xd9 h3 y2d2 ff2 fs0 fc0 sc0 ls0 ws0">Pr[<span class="ff3">X </span>=<span class="_ _7"> </span><span class="ff3">k<span class="_ _15"></span></span><span class="ls1">]=</span></div><div class="t m0 x11b h6 y2d3 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xfa h3 y2d4 ff3 fs0 fc0 sc0 ls0 ws0">n</div><div class="t m0 xfa h3 y2d5 ff3 fs0 fc0 sc0 ls0 ws0">k</div><div class="t m0 x25 h6 y2d3 ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 xb h3 y2d2 ff3 fs0 fc0 sc0 ls0 ws0">p</div><div class="t m0 x15e h5 y2d6 ff6 fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 x12e h3 y2d7 ff3 fs0 fc0 sc0 ls0 ws0">q</div><div class="t m0 xc h5 y2d6 ff6 fs1 fc0 sc0 ls0 ws0">n<span class="ff8">−</span>k</div><div class="t m0 xb3 h4 y2d7 ff3 fs0 fc0 sc0 lsd ws0">,q<span class="_ _40"></span><span class="ff2 ls1">=1<span class="_ _5"></span><span class="ff4 ls0">−<span class="_ _8"></span><span class="ff3">p,</span></span></span></div><div class="t m0 xca h3 ybf ff2 fs0 fc0 sc0 ls0 ws0">E</div><div class="t m0 x189 h3 y2d8 ff2 fs0 fc0 sc0 ls0 ws0">[<span class="ff3">X<span class="_ _15"></span></span><span class="ls1">]=</span></div><div class="t m0 xeb h5 y2d9 ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x11b h6 y2da ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x11b h5 y2db ff6 fs1 fc0 sc0 ls0 ws0">k<span class="ff5">=1</span></div><div class="t m0 x27 h3 y2dc ff3 fs0 fc0 sc0 ls0 ws0">k</div><div class="t m0 x71 h6 y2dd ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x58 h3 ybe ff3 fs0 fc0 sc0 ls0 ws0">n</div><div class="t m0 x58 h3 y189 ff3 fs0 fc0 sc0 ls0 ws0">k</div><div class="t m0 xc h6 y2dd ff7 fs0 fc0 sc0 ls0 ws0"></div><div class="t m0 x46 h3 y2dc ff3 fs0 fc0 sc0 ls0 ws0">p</div><div class="t m0 x73 h5 y186 ff6 fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 xb3 h3 y2dc ff3 fs0 fc0 sc0 ls0 ws0">q</div><div class="t m0 x2b h5 y186 ff6 fs1 fc0 sc0 ls0 ws0">n<span class="ff8">−</span>k</div><div class="t m0 x48 h3 y2dc ff2 fs0 fc0 sc0 ls0 ws0">=<span class="_ _7"> </span><span class="ff3">np.</span></div><div class="t m0 x13d h3 y29f ff2 fs0 fc0 sc0 ls0 ws0">P<span class="_ _5"></span>oisson distribution:</div><div class="t m0 x8e h3 y1db ff2 fs0 fc0 sc0 ls0 ws0">Pr[<span class="ff3">X </span>=<span class="_ _7"> </span><span class="ff3">k<span class="_ _15"></span></span><span class="ls1">]=</span></div><div class="t m0 x44 h3 y18e ff3 fs0 fc0 sc0 ls0 ws0">e</div><div class="t m0 x71 h5 y191 ff8 fs1 fc0 sc0 ls0 ws0">−<span class="ff6">λ</span></div><div class="t m0 x9c h3 y18e ff3 fs0 fc0 sc0 ls0 ws0">λ</div><div class="t m0 xef h5 y191 ff6 fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 x9b h3 y216 ff3 fs0 fc0 sc0 ls0 ws0">k<span class="_ _3"></span><span class="ff2">!</span></div><div class="t m0 x29 h3 y1db ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 xaa h3 y2de ff2 fs0 fc0 sc0 ls0 ws0">E</div><div class="t m0 x2a h3 y1db ff2 fs0 fc0 sc0 ls0 ws0">[<span class="ff3">X<span class="_ _15"></span></span><span class="ls1">]=</span><span class="ff3">λ.</span></div><div class="t m0 x13d h3 y2df ff2 fs0 fc0 sc0 ls0 ws0">Normal (Gaussian) distribution:</div><div class="t m0 x1 h3 y199 ff3 fs0 fc0 sc0 ls0 ws0">p<span class="ff2">(</span>x<span class="ff2 ls1">)=</span></div><div class="t m0 x15d h3 y217 ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 x8 h4 y199 ff4 fs0 fc0 sc0 ls0 ws0">√</div><div class="t m0 x150 h3 y2e0 ff2 fs0 fc0 sc0 ls0 ws0">2<span class="ff3 ls10">πσ</span></div><div class="t m0 xeb h3 y199 ff3 fs0 fc0 sc0 ls0 ws0">e</div><div class="t m0 x153 h5 ye2 ff8 fs1 fc0 sc0 ls0 ws0">−<span class="ff5">(<span class="ff6">x</span></span>−<span class="ff6">µ<span class="ff5">)</span></span></div><div class="t m0 x18a h7 y2e1 ffc fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x72 h5 ye2 ff6 fs1 fc0 sc0 ls0 ws0">/<span class="ff5">2</span>σ</div><div class="t m0 xd h7 y2e1 ffc fs2 fc0 sc0 ls0 ws0">2</div><div class="t m0 x47 h3 y199 ff3 fs0 fc0 sc0 ls0 ws0">,</div><div class="t m0 x126 h3 ye0 ff2 fs0 fc0 sc0 ls0 ws0">E</div><div class="t m0 xb4 h3 y199 ff2 fs0 fc0 sc0 ls0 ws0">[<span class="ff3">X<span class="_ _15"></span></span><span class="ls1">]=</span><span class="ff3">µ.</span></div><div class="t m0 x13d h3 y2e2 ff2 fs0 fc0 sc0 ls0 ws0">The<span class="_ _34"> </span>“coup<span class="_ _3"></span>on<span class="_ _34"> </span>collector”:<span class="_ _21"> </span>W<span class="_ _5"></span>e<span class="_ _0"> </span>are<span class="_ _34"> </span>given<span class="_ _0"> </span>a</div><div class="t m0 x13d h3 y2e3 ff2 fs0 fc0 sc0 ls0 ws0">random coupon eac<span class="_ _5"></span>h<span class="_ _7"> </span>day<span class="_ _3d"></span>, and<span class="_ _7"> </span>there are<span class="_ _7"> </span><span class="ff3">n</span></div><div class="t m0 x13d h3 y2e4 ff2 fs0 fc0 sc0 ls0 ws0">differen<span class="_ _5"></span>t<span class="_ _0"> </span>t<span class="_ _5"></span>yp<span class="_ _3"></span>es<span class="_ _0"> </span>of<span class="_ _0"> </span>coup<span class="_ _3"></span>ons.<span class="_ _11"> </span>The<span class="_ _0"> </span>distribu-</div><div class="t m0 x13d h3 y2e5 ff2 fs0 fc0 sc0 ls0 ws0">tion of coupons is uniform.<span class="_ _1e"> </span>The<span class="_ _7"> </span>exp<span class="_ _3"></span>ected</div><div class="t m0 x13d h3 y2e6 ff2 fs0 fc0 sc0 ls0 ws0">n<span class="_ _5"></span>umber<span class="_ _0"> </span>of<span class="_ _0"> </span>da<span class="_ _5"></span>ys<span class="_ _0"> </span>to<span class="_ _0"> </span>pass<span class="_ _0"> </span>before<span class="_ _0"> </span>we<span class="_ _0"> </span>to<span class="_ _0"> </span>col-</div><div class="t m0 x13d h3 y2e7 ff2 fs0 fc0 sc0 ls0 ws0">lect all <span class="ff3">n </span><span class="lsc">ty<span class="_ _3"></span>p<span class="_ _15"></span>e<span class="_ _3"></span>s<span class="_ _0"> </span>is</span></div><div class="t m0 xa h3 y109 ff3 fs0 fc0 sc0 ls0 ws0">nH</div><div class="t m0 x9c h5 y2e8 ff6 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 xef h3 y109 ff3 fs0 fc0 sc0 ls0 ws0">.</div><div class="t m0 x80 h3 y2e9 ff2 fs0 fc0 sc0 ls0 ws0">29<span class="_ _4b"> </span>536,870,912<span class="_ _4c"> </span>109</div><div class="t m0 x80 h3 y297 ff2 fs0 fc0 sc0 ls0 ws0">30<span class="_ _d"> </span>1,073,741,824<span class="_ _4d"> </span>113</div><div class="t m0 x80 h3 y2ea ff2 fs0 fc0 sc0 ls0 ws0">31<span class="_ _d"> </span>2,147,483,648<span class="_ _4d"> </span>127</div><div class="t m0 x80 h3 y29c ff2 fs0 fc0 sc0 ls0 ws0">32<span class="_ _d"> </span>4,294,967,296<span class="_ _4d"> </span>131</div><div class="t m0 x140 h3 y2eb ff2 fs0 fc0 sc0 ls0 ws0">P<span class="_ _5"></span>ascal’s T<span class="_ _5"></span>riangle</div><div class="t m0 xf2 h3 y2ec ff2 fs0 fc0 sc0 ls0 ws0">1</div><div class="t m0 xe5 h3 y2a5 ff2 fs0 fc0 sc0 ls13 ws0">11</div><div class="t m0 xf1 h3 y2ed ff2 fs0 fc0 sc0 ls13 ws0">121</div><div class="t m0 xe3 h3 y2ee ff2 fs0 fc0 sc0 ls13 ws0">1331</div><div class="t m0 x18b h3 y2ae ff2 fs0 fc0 sc0 ls13 ws0">14641</div><div class="t m0 x18c h3 yf3 ff2 fs0 fc0 sc0 ls0 ws0">1 5 10 10 5 1</div><div class="t m0 x87 h3 y2ba ff2 fs0 fc0 sc0 ls0 ws0">1 6 15 20 15 6 1</div><div class="t m0 x18d h3 y19e ff2 fs0 fc0 sc0 ls0 ws0">1 7 21 35 35 21 7 1</div><div class="t m0 x8f h3 y2ef ff2 fs0 fc0 sc0 ls0 ws0">1 8 28 56 70 56 28 8 1</div><div class="t m0 x84 h3 y2f0 ff2 fs0 fc0 sc0 ls0 ws0">1 9 36 84 126 126 84 36 9 1</div><div class="t m0 x114 h3 y2c0 ff2 fs0 fc0 sc0 ls0 ws0">1 10 45 120 210 252 210 120 45 10 1</div></div><div class="pi" data-data='{"ctm":[1.673203,0.000000,0.000000,1.673203,0.000000,0.000000]}'></div></div></div>