A component has a failure rate of $\lambda$ (i.e. the probability of failure in period ∆t is λ∆t).
Show that the probability that the component will have failed by time t is $F(t) = 1 − e^{−\lambda t}$
Derive the mean and variance of the time to failure.
I am stuck in the last part of the question where we need to find the variance. I know that
$$\mathbb E(t^2)=\int_0^\infty t^2\cdot \lambda e^{−\lambda t} \,dt $$
How do I convert the above integral into Laplace so that it is easy to solve?
The answer for the variance is $\frac1{\lambda^2}$.