Double integral setup - Uniform distribution

Question

I am currently trying to understand a specific component of a probability problem involving setting up the proper bounds on a double integral.

In the problem, $X_1$ and $X_2$ are independent Uniform $(0,1)$ random variables. Thus, $f_{X_1,X_2}(x_1, x_2) = f_{X_1} \cdot f_{X_2} = $ 1 (if $X_1$ and $X_2$ are both between 0 and 1).

I am asked to find $$P(X_1 > C-X_2) = \iint 1 \, dx_1\,dx_2\,.$$ My confusion lies in how to set up the bounds of integration for this situation. I know that the range of $x_2$ values goes from 0 to 1, but how does one find the proper bounds for the inner integral. It's been a long time since I took a Calc 2/Calc 3 course and my double integral skills are extremely poor. Any help is greatly appreciated.

Answer 1

Sketching the region helps. I will use $X, Y$ instead of $X_1, X_2$.

If $ ~0 \lt C \lt 1$,

$P(X \gt C - Y) =1 - P(X \lt C - Y)$

$ \displaystyle = 1 - \int_0^C \int_0^{C-y} ~ dx ~ dy = 1 - \frac{C^2}{2}$

If $ ~1 \lt C \lt 2$,

$P(X \gt C - Y) = \displaystyle \int_{C-1}^1 \int_{C-y}^1 ~ dx ~ dy = \frac{(C-2)^2}{2}$

Answer 2

It is $$\int_0^{C} \int_{C-x_2}^{1}dx_1dx_2$$ if $0 <C<1$, $$\int_{C-1}^{C} \int_{C-x_2}^{1}dx_1dx_2$$ if $1 \leq C<2$,

$0$ if $C \geq 2$ and

$1$ if $C \leq 0$.