Calculating probability density function I am having difficulty solving this problem.

Given this distribution
  $$
\begin{align*}
 f_X(x) &= \begin{cases}e^{-x} &,& x\geq 0 \\
  0 &,& x<0\end{cases} \\[1ex]
 f_Y(y) &= \begin{cases}2e^{-2y} &,& y\geq 0 \\
  0 &,& y<0\end{cases}
\end{align*}
$$
  Calculate $\mathsf P(X<3Y)$

I have attempted the following, however I am unsure if my methodology is correct.
$$
 \mathsf P(X<3Y)= \int_0^\infty \left[\int_0^{3y} f_X(x)f_Y(y) dx\right] dy
$$
 A: Your random variable $X$ has an exponential distribution with rate $\lambda = 1,$ and it has mean $E(X) = 1.$
Your random variable $Y$ has an exponential distribution with rate $\lambda = 2,$ and it has mean $E(Y) = 1/2.$
One approach to finding $P(X < 3Y),$ for $X$ and $Y$ independent, is to write their joint distribution, and then integrate to find the probability.
This specific probability can be approximated very accurately using simulation.
That is the approach I show below, using R statistical software to get
100,000 simulated realizations of $(X,Y).$ The result of the simulation is that $P(X < 3Y) = .60,$
correct to about two decimal places.
m = 10^5;  x = rexp(m, 1);  y = rexp(m, 2)
mean(x);  mean(y);  event = (x < 3*y);  mean(event)
## 1.003198   # aprx E(X) = 1
## 0.5021866  # aprx E(Y) = 1/2
## 0.60261    # aprx P(X < 3Y) = .6

The scatterplot below shows the 100,000 simulated points.
Almost exactly 60% of the points are green, corresponding to the
event $\{X < 3Y\}.$ (If you are solving this by integration, the
green region is the region over which you need to integrate
the joint density function.)

