I am working on a homework problem below from Pitman's Probability book:
Suppose the failure rate is $\lambda$($t$) = $at$ $+$ $b$ for $t$ $\geq 0$.
The problem asks to find the formula for the survival function, the density function and the mean lifetime, all which I have done. The final piece if finding the SD of the lifetime. I know that I want to find the $\operatorname{Var}(T)$ since the standard deviation is the square root of the variance. This would involve finding the expectation of the square of $T$ wherein lies my problem.
The expectation of the square of $T$ is given by the following integral $2\int_0^\infty tG(t) \,dt$. With my acquired survival function $G(t) = \exp(-at^2/2+bt$). I'm trying to evaluate
$ 2\int_0^\infty t \cdot \exp(-at^2/2+bt)\, dt. $
With this seemingly dead end, I attempted to find $E(T^2)$ using the density function. That is, $E(T^2) = \int_0^\infty t^2 \cdot f(t)\,dt$ where $f(t)=(at+b) \cdot \exp(-at^2/2+bt)$. This integral, in my observation, would be easy to evaluate if the exponent matched the product term, but I can't seem to find a way to make this so without over complicating the problem.
Any help would be greatly appreciated. I did the best that I could in conveying my progress so to affirm that I am not looking for the answer, but a much needed hint/guide. Thanks