In this question we have that $G \sim \operatorname{Exp}(0.3)$ and $M \sim \operatorname{Exp}(0.6)$ where $G$ and $M$ are independent of each other. I need to find the joint density function and then find $P(G>2M)$.
So far I have that the joint density function is
\begin{align}
& f_{GM}(g,m)=f_G(g)f_M(m) \\[10pt]
= {} & (0.3e^{-0.3g})(0.6e^{-0.6m}) \\[10pt]
= {} & 0.18e^{-0.3g-0.6m} \text{(and $0$ otherwise??)}
\end{align}
If this is correct then what is the domain of the density function? If it is $h,m>0$ then I'm not sure what the limits for each of my integrals are. I believe to find $P(G>2M)$ I need to solve a double integral like this:
$$\iint 0.18e^{-0.3g-0.6m} \, dg \, dm$$
…but I don't know how to find the limits of integration. Thanks for any input.
Best Answer
It is the cartesian product of the support domains of the marginals. $\{(g,m)\in\Bbb R^2: g\geq 0, m\geq 0\}$
$$f_{G,M}(g,m) =0.18 \mathsf e^{-0.3g-0.6m}~\mathbf 1_{g\geq 0, m\geq 0}$$
And of course the domain of the integration for the event of $G\geqslant 2M$ will be $\{(g,m)\in\Bbb R^2: g\geq 2m, m>0\}$ $$\mathsf P(G\geqslant 2M)=\int_0^\infty \int_{2m}^\infty 0.18\mathsf e^{-0.3g-0.6m}~\mathsf d g~\mathsf d m$$