Integration by parts with continuous random variables

Question

Show that for any continuous positive random variable $X$ with $F(x) = F_X(x)$

we have $$EX =\int_{0}^{\infty} (1 − F(x)) \text{ d}x\text{.}$$

[Hint: use integration by parts on $(1 − F(x)) · 1$.]

I have no idea how to use the hint, could someone help for this? Thanks

If you assume that $X$ has a density (recall that the density is the derivative of the CDF), then $E[X] = \int_0^\infty x F'(x) dx$. — Fnacool, Mar 24 '16 at 14:17
Possible duplicate of Expected value as integral of survival function — Clarinetist, Mar 24 '16 at 14:18
Are you forced to use integration by parts? If you are not, the question is a multiplicate of the site... — Did, Mar 24 '16 at 14:50

score 1 · Accepted Answer · answered Mar 24 '16 at 15:10

1

Hint:

$$\int_{(0,b]}x^aF(x)dx=-b^a(1-F(b))+a\int_{(0,b]}x^{a-1}(1-F(x))dx$$

If you need more hints feel free to ask.

answered Mar 24 '16 at 15:10

MorphhproM

1 Answers1