Let Xi I 1 n be independent random variables having the Un

Let 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_3}. Show that the distribution of Y_n as n rightarrow infinity is the Exponential distribution with lambda = - 1.

Solution

as we can see Yn is n * min )x1, x2, xn )

and X1, X2, ... Xn are uniform distribution between 0 and 1 so the pdf is

f(X1) = 1

mulitply by n that is N the number of independent random variables

so as n is big ( infiniti)

we will have that our density function of Y will be a exponential distribution with landha = -1