pe 2. Let X be a random variable whose probability mass function is given by P(X = x) = p(1 – p)*-1 for x = 1,2,..., (1)

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Pe 2 Let X Be A Random Variable Whose Probability Mass Function Is Given By P X X P 1 P 1 For X 1 2 1 1 (150.36 KiB) Viewed 41 times

pe 2. Let X be a random variable whose probability mass function is given by P(X = x) = p(1 – p)*-1 for x = 1,2,..., (1) where 0 <p<1 is the parameter of the distribution. (a) For 0 <- log(1 - p), define moment generating function of X by Mx(O) = E [eºx]. Show that Mx(0) = (2) (1 -(1-pe) Hint: Use the following facts: for al < 1, 2X=0 af = 1-a. (b) Using (2), determine the mean and variance of X. (c) For a given positive integer x, show that P(X > x) = (1 - p)* (d) Set p=1/n in (3). Show that X/n has the limiting exponential distribution with intensity one as n + 0, i.e., P(X/n >x/n) + e-x/n as n. as n → Hint: Use the following fact: (1-x/n)" →e-*, as n+. (3) (17 marks)