How can I determine an expected value of a variable that depends on a probability distribution function?

Question

The distribution of downtime $T$ for breakdowns of a system is given by

$$f(t) = (a^2)te^{-at} \text{ for }t > 0$$

where $a$ is a positive constant.

The cost of downtime derived from the distruption resulting from breakdowns rises exponentially with $T$:

$$\text{cost factor} = h(T) = e^{bT}$$

Show that the expected cost factor for downtime is $$\left(\frac{a}{a-b}\right)^2$$, provided that $a > b$.

How can I show this? I know how to calculate the mean of the pdf but how do I combine it with the other function?

Answer 1

For any function $g$, the expected value $\mathbb{E}[g(Y)] = \int g(y)f(y)$ where $f(y)$ is the probability density function.

Answer 2

The expected cost factor is: $$\int_0^{+\infty} h(t) f(t) dt=\int_0^{+\infty} e^{bt} a^2 t e^{-at} dt=a^2 \int_0^{+\infty} t e^{-(a-b)t} dt $$ and (for $a>b$): $$\int_0^{+\infty} t e^{-(a-b)t} dt=\left(\frac{1}{a-b}\right)^2$$