By Peter J. Cameron
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Extra info for Notes on Probability [Lecture notes]
Example text
Then (a) [GX (x)]x=1 = 1; (b) E(X) = (c) Var(X) = d dx GX (x) x=1 ; d2 G (x) + E(X) − E(X)2 . dx2 X x=1 Part (a) is just the statement that probabilities add up to 1: when we substitute x = 1 in the power series for GX (x) we just get ∑ pk . For part (b), when we differentiate the series term-by-term (you will learn later in Analysis that this is OK), we get d GX (x) = ∑ kpk xk−1 . dx Now putting x = 1 in this series we get ∑ kpk = E(X). For part (c), differentiating twice gives d2 GX (x) = ∑ k(k − 1)pk xk−2 .
Let’s check that these probabilities add up to one. We get λk ∑ k=0 k! ∞ e−λ = eλ · e−λ = 1, since the expression in brackets is the sum of the exponential series. By analogy with what happened for the binomial and geometric random variables, you might have expected that this random variable would be called ‘exponential’. Unfortunately, this name has been given to a closely-related continuous random variable which we will meet later. However, if you speak a little French, you might use as a mnemonic the fact that if I go fishing, and the fish are biting at the rate of λ per hour on average, then the number of fish I will catch in the next hour is a Poisson(λ) random variable.
Pi−1 ’, then P(Ai | A1 , . . 6. 5, 1 2 P(A1 ∩ · · · ∩ Ai ) = (1 − 365 )(1 − 365 ) · · · (1 − i−1 365 ). Call this number qi ; it is the probability that all of the people p1 , . . , pi have their birthdays on different days. The numbers qi decrease, since at each step we multiply by a factor less than 1. 5, that is, n is the smallest number of people for which the probability that they all have different birthdays is less than 1/2, that is, the probability of at least one coincidence is greater than 1/2.
Notes on Probability [Lecture notes] by Peter J. Cameron
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