If $X$ ~ $U[0, 4]$ and $Y$~$[0, 7]$ find the probability X value is greater than Y value

Question

Suppose $X$ and $Y$ are continuous uniform random variables. If $X$ ~ $U[0, 4]$ and $Y$~$[0, 7]$ find the probability that a random $X$ value is greater than a random $Y$ value.

Answer 1

We need to assume independence. Draw the rectangle with corners $(0,0)$, $(4,0)$, $(4,7)$, and $(0,7)$. Draw the line $y=x$.

We want to find the probability that the pair $(X,Y)$ lands below the line $y=x$. So we want to find the probability $(X,Y)$ lands in a certain triangle. That probability is the area of the triangle, divided by the area of the whole rectangle.

I am confident you can take care of the rest of the calculation.

Remark: We can also find the answer by integrating. Let $T$ be the triangle described above. The joint density function of $X$ and $Y$ is $\frac{1}{28}$ in the rectangle, and $0$ elsewhere. So the required probability is $$\iint_T \frac{1}{28}\,dy \,dx.$$ The integral is not difficult to evaluate.