Page 197, Q6

  1. Let X1, X2, . . . , Xn be n mutually independent random variables, each of which is uniformly distributed on the integers from 1 to k. Let Y denote the minimum of the Xi’s. Find the distribution of Y .

This is an order statistic problem

\[ Given\ CDF\ of\ 1st\ order\ statistic\ is\ F_1(x)=1-(1-F_x(x))^n \] \[F_x(x) = \frac{y}{k}\] \[1-F_x(x) = \frac{k}{k}-\frac{y}{k} = \frac{k-y}{k}\]

\[ P(Y=y) = P(Y\leqslant y) - P(Y\leqslant y-1) \] \[ P(Y=y) = (1-(\frac{k-y}{k})^n)- (1-(\frac{k-y+1}{k})^n)\] \[ P(Y=y) = -(\frac{k-y}{k})^n + (\frac{k-y+1}{k})^n\] \[ P(Y=y) = \frac{(k-y+1)^n-(k-y)^n}{k^n}\]