Lel X_1,...X_n be independent and identically distributed continuous random vari

Question

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.

Explanation / Answer

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

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