Question: 7. Prove that G, the distribution function of [X(1) + X(n)] 2, the midrange of a random sample of size n from a population with

7. Prove that G, the distribution function of [X(1) + X(n)]

2, the midrange of a random sample of size n from a population with continuous distribution function F and probability density function

f, is given by

G(t) = n - n [* [F(2t x) F(x)]" f(x)dx.

Hint: Use Theorem 9.6 to find f1n; then integrate over the region x + y ≤ 2t and x ≤ y.

G(t) = n - n [* [F(2t x) F(x)]" f(x)dx.

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