conditional probability density function

Question

If the joint probability density function for the waiting times $W_1$ and $W_2$ is given by:

$f(w_1,w_2)=\lambda^2$ $exp(-\lambda w_2)$ for $0<w_1<w_2$.

How would I determine the conditional probability density function for $W_1$, given that $W_2=w_2$?

Answer 1

Hint: The conditional probability density function $g(w_1|W_2=w_2)=\dfrac{f(w_1, w_2)}{h_{W_2}(w_2)}$. But the marginal distribution $h_{W_2}(w_2)=\int_{-\infty}^\infty f(w_1, w_2) dw_1=\int_0^{w_2} \lambda^2 e^{-\lambda w_2}dw_1$.