Finding Expectation from the inverse CDF.

[Dr. Rostov]

Homework Statement

http://209.85.48.12/3560/8/upload/p2791776.jpg

Homework Equations

The most relevant identity to the part that I'm confused about is the following identity: for any cumulative distribution function F, with the inverse function F^-1, if U has uniform (0,1) distribution, then F^-1(U) has cdf F. Also useful: E(X) is the integral from -$$\infty$$ to $$\infty$$ of x * f(x), where f(x) = the probability distribution function of the distribution.

The Attempt at a Solution

The identity given for E(X) of a CDF makes perfect sense to me, and deducing the discrete corollary to the theorem makes sense too. Part C isn't anything I need help on, either; I've already used the formula to get it. But though I understand that these all make sense, I'm really just kind of confused about what they're asking in part A. What do they mean by using X=F^-1(U) to show that E(X) can be interpreted as the shaded area above the CDF of X? Basically, what's the convention for integrating an inverse function representing a function that you don't know? I'm not looking for any coddling -- I'm just really rather confused by this problem and would like a push in the right direction to figure out what's up here.

 

[Billy Bob]

what's the convention for integrating an inverse function representing a function that you don't know?

Not so much that point of view, but more like the reverse. The phrase "using X=F^-1(U)" means use this change of variables to change the integral for E[X] into an integral in terms of F^-1(u) and du.