10_DiscreteRVs-2_Expectation_packed

C k 1 p k 1 p k 1 d useful fact k 1

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Unformatted text preview: − p ) ⎬ ⎣ ⎦ dp dp ⎩ k 1 ⎩ k =1 ⎭ = ⎭ (1− p )2 / p −1 / p ⎫1 ⎫ 1 2p 1 2 1 d2 ⎧ 1 d⎧ 1 = p 2 ⎨ − 2 + p⎬ − = p ⎨− 2 + 1⎬ − = 3 − = 2 − dp ⎩ p dp ⎩ p pp p ⎭p ⎭pp Ilya Pollak Variance of a geometric random variable 2 1 ⎛1⎞ 11 2 2 var(C ) = E [C ] − ( E [C ]) = 2 − − ⎜ ⎟ = 2 − p p ⎝ p⎠ p p 2 If p = 1/200, 000, the standard deviation is 200, 000 2 − 200, 000 ≈ 200, 000 Thus, your actual number candy bars until first success may be quite far from the mean! Ilya Pollak...
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