Joint probability density for non-identical Uniform random variables

Question

Let $~X,~ Y~$ be uniform on $~[0, 3] × [2, 4]~$. Find $~P(X + Y ≤ 5)~$ and
$~X~$ and $~Y~$ are independent.

My approach: Using convolution formula.

Difficulty I am facing: Understanding the limits of the integration

Note: I have explored almost every answer about joint PDF and CDF for uniform RVs but most of those has iid random variables. but in this case, $~X~$ and $~Y~$ are independent but not identical.

Answer 1

Guide:

• You don't really need to use integration. You have think in terms of ratio of areas.

• To figure out the limit of integration, refer to the picture and try to describe the corresponding region.

• It might be simpler to deal with the complement.

Answer 2

Siong Thye Goh's answer is excellent.

But if you really feel you need it, the integral of interest would be:

$$\begin{align}\mathsf P(Y\leqslant5-X) &=\int_0^3\int_{2}^{\min\{4,5-x\}}\frac 1{6}~\mathrm d y~\mathrm d x\\[2ex]& =\int_0^1\int_2^4 \frac16~\mathrm d y~\mathrm d x+\int_1^3\int_2^{5-x}\frac 16~\mathrm d y~\mathrm d x\end{align}$$

Or by complements$$\begin{align}\mathsf P(X+Y\leq 5) &=1-\mathsf P(5-Y<X)\\[2ex]&=1-\int_2^4\int_{5-y}^{3}\frac 16~\mathrm d x~\mathrm d y\end{align}$$