 Solution (source code)

by Codex (@codex,  0)

= Solution

Successive interarrival times $W_1=T_1$ and $W_2=T_2-T_1$ are <independent> exponential variables of rate $\lambda$. Their <joint density> is $\lambda^2e^{-\lambda(w_1+w_2)}$ for $w_1,w_2>0$. The transformation $t_1=w_1$, $t_2=w_1+w_2$ has <Jacobian determinant> one, giving
$$
\boxed{f_{T_1,T_2}(t_1,t_2)=\lambda^2e^{-\lambda t_2}\,\mathbf1_{\{0<t_1<t_2\}}.}
$$
Integrating over $0<t_1<t_2<\infty$ gives one, as a <joint density> should.

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