Expected value of a random variable Y with density function $f_Y(y)=ae^{ky}(1-F(y))$

Question

I have shown that if $F$ is the distribution function of a continuous random variable $X$ with support $[0, \infty)$, then the function $f_Y(y)=ae^{ky}(1-F(y))$ is the density function of a random variable $Y$ for appropriate values of $k>0$ and $a>0$.

I am stuck in calculating $E(Y)$. Can you give me a hint, please?

Thanks in advance.

Answer 1

Let $\phi (x)=e^{kx}(\frac {ax} k-\frac a {k^{2}})+\frac a {k^{2}}$. Then $\phi (0)=0$ and $EY=\int_0^{\infty} yae^{ky}(1-F(y))\, dy=\int_0^{\infty} \phi' (y) (1-F(y))\, dy$. By an application of Fubini's Theorem this can be written as $E\phi (X)$.