Question

The joint density function for a pair of random variables x and y is given f(x,y)= {cx(1+y) if 0≤x≤1, 0≤y≤2 0 otherwise find the value of the constant
c. find p(x+y≤1)

Answer 1

$f_{X,Y}(x,y)=\begin{cases}cx(1+y)&\text{for }0\le x\le1,0\le y\le2\\0&\text{otherwise}\end{cases}$


For $f$ to be a proper density function, we need to have the integral over its support $\mathcal S$ to equal 1.


$\displaystyle\iint_{\mathcal S}f_{X,Y}(x,y)\,\mathrm dx\,\mathrm dy=\int_{y=0}^{y=2}\int_{x=0}^{x=1}cx(1+y)\,\mathrm dx\,\mathrm dy=2c=1$


$\implies c=\dfrac12$


Now,


$\mathbb P(X+Y\le1)=\mathbb P(X\le1-Y)=\displaystyle\int_{y=0}^{y=2}\int_{x=0}^{x=1-y}\frac{x(1+y)}2\,\mathrm dx\,\mathrm dy=\frac13$