Lel X1Xn be independent and identically distributed continuo

Lel X_1,...X_n be independent and identically distributed continuous random variables having cumulative distribution function F(x) and and density function f(x). The quantity M = [X_11 + X_11]/2. defined to be the average of the smallest and largest values in X_1...X_11 is called the \"midrange\" of the sequence. Show that M has the cumulative distribution function Also calculate this function when X, are (a) independent identical Exponential random variables with parameter lambda. and (b) independent identical Uniform (O.I) random variables.

Solution

a)

as you can see FM(M) has a f(x) multiply by the distribution of F

since we know that f(x) is a probability distribution (mass) we knwo that if we

multiply that for Fm we willhave a cumulative distribution function