Compute the expectation given cdf.

Question

Find the expectation of $X$ given that:

$F_X(x) = 0$ if $x < 0$;

$F_X(x) = 2x-x^2$ if $0\le x \le 1$;

$F_X(x) = 1$ if $x \ge 1$

I don't know what to do. Can you help me?

Answer 1

Two possibilities:

• check this is a continuous distribution, differentiate to find the density $f(x)=F'(x)$ and then find $\int\limits_{-\infty}^\infty x\, f(x)\, dx$

• check this is a non-negative distribution, find the survival function $S(x)=1-F(x)$ and then find $\int\limits_0^\infty S(x)\, dx$