Lei Xi i 1 n be independent random variables having the Un

Lei X_i, i = 1, ..., n be independent random variables having the Uniform distribution between 0 and 1. Let us define Y_n = n Min {X_1, X_2, ...,X_n}. Show that the distribution of Y_n as n rightarrow infinity is the Exponential distribution with lambda = - 1.

Solution

Problem 6

as we can see Xn ---> f(X) = 1

and

Yn = n * f(X)

since n is increase we will have a exponential distribution with landha = -1

Yn = -1 * f(x)