Solution
= Solution
The <Poisson interarrival times> of a homogeneous <Poisson process> are <independent and identically distributed> with the <exponential distribution> of rate $\lambda$. Consequently $Y_1,Y_2$ are <independent random variables> and their <joint probability density> is
$$
\boxed{f_{Y_1,Y_2}(y_1,y_2)=\lambda^2e^{-\lambda(y_1+y_2)}\mathbf1_{\{y_1>0,\ y_2>0\}}.}
$$
Here $Y_2$ is the second interarrival duration, rather than the time of the second arrival.