# Difficult Definite Integral Question

1. Find the expected value of the largest order statistic in a random sample of size 4 from the standard normal distribution.

2. Homework Equations

E(X(4,4))=4∫xf(x)(F(x))^3dx, (from minus infinite to plus infinite), where f(x) is the probability density function of standard normal distribution, F(x) is the culmulative density function of standard normal distribution.

3. The Attempt at a Solution
To evaluate the definite integral in the above equation, my approach is to use integral by part, the simplifed equation is as follow:

E(X(4,4)) =-24∫xf(x)F(x)dx+24∫(xf(x)F(x))^2dx (from minus infinite to plus infinite)

On the right side of the above equation, we can evaluate the first definite integral by changing to polar coordinates, however, it is very difficult for me to evalutate the second definite integral on the right side of the above equation using integral by parts.

4.Question
How to evaluate the second definite integral ( from minus infinite to plus infinite). or What is the trick of evaluating this definite integral ? Please help!

Attahed is the my detailed computation process for your reference. I am not sure whether or not my approach to evaluating this definite integral is correct.

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