# Thread: Relationship with the expectation value

1. ## Relationship with the expectation value

In a recent homework question I was asked to find the followings:

$\displaystyle E(X)$and$\displaystyle \int_{0}^{\infty}[1-F_X(x)]dx$
$\displaystyle F_X(x)=P(X\leq x)=\left\{\begin{matrix} (1-1/x^2) \ \ if \ \ x\geq 1\\ 0 \ \ elsewhere \end{matrix}\right.$

This was easy, I found both to be 1. What bothered me was an remark that the prof made at the end of the question: he wrote "This is not a coincidence."

Is this always true? Are there any proofs? Does it depend of $\displaystyle F_X(x)$? What has the expectation got to do with $\displaystyle \int_{0}^{\infty}[1-F_X(x)]dx$? I though the expectation is just like taking an average.

2. Originally Posted by synclastica_86
In a recent homework question I was asked to find the followings:

$\displaystyle E(X)$and$\displaystyle \int_{0}^{\infty}[1-F_X(x)]dx$
$\displaystyle F_X(x)=P(X\leq x)=\left\{\begin{matrix} (1-1/x^2) \ \ if \ \ x\geq 1\\ 0 \ \ elsewhere \end{matrix}\right.$

This was easy, I found both to be 1. What bothered me was an remark that the prof made at the end of the question: he wrote "This is not a coincidence."

Is this always true? Are there any proofs? Does it depend of $\displaystyle F_X(x)$? What has the expectation got to do with $\displaystyle \int_{0}^{\infty}[1-F_X(x)]dx$? I though the expectation is just like taking an average.