# Finding probability of the given joint pdf

I am given following joint pdf:

$$f(x,y)=\frac{2}{3}(x+2y)$$ $$0\leq x \leq1, 0\leq y \leq1$$

Now I need to find the $$P[X\leq0.5,Y\leq0.8]$$.

My question is that will I double integrate the expression from 0 till respective limits like this:

$$\int_{0}^{0.8}\int_{0}^{0.5} \frac{2}{3}(x+2y) \,dx\,dy$$

• Yes, that is okay. Dec 16 '18 at 16:36
• There is no confusion. For any $a,b\in(0,1)$, \begin{align} P(X\le a, Y\le b)&=E\left[\mathbf1_{X\le a,Y\le b}\right] \\&=\iint \mathbf1_{x\le a,y\le b }\,f(x,y)\,\mathrm{d}x\,\mathrm{d}y \\&=\frac{2}{3}\iint\mathbf1_{x\le a,y\le b }(x+2y)\mathbf1_{0<x<1,0<y<1}\,\mathrm{d}x\,\mathrm{d}y \\&=\frac{2}{3}\int_0^b \int_0^a (x+2y)\,\mathrm{d}x\,\mathrm{d}y \end{align} Dec 16 '18 at 16:42