Question

The random variables X and Y have the joint PMF pX,Y(x,y)={c⋅(x+y)2,0,if x∈{1,2,4} and y∈{1,3},otherwise. All answers in this problem should be numerical. Find the value of the constant c .

Answer 1

In order for $p_{X,Y}(x,y)$ to be a valid PMF, its integral over the distribution's support must be equal to 1.

$p_{X,Y}(x,y)=\begin{cases}\dfrac{c(x+y)}2&\text{for }x\in\{1,2,4\}\text{ and }y\in\{1,3\}\\\\0&\text{otherwise}\end{cases}$

There are 3*2 = 6 possible outcomes for this distribution, so that

$\displaystyle\sum_{x,y}p_{X,Y}(x,y)=\sum_{x\in\{1,2,4\}}\sum_{y\in\{1,3\}}\frac{c(x+y)}2=1$

$1=\displaystyle\frac c2\sum_{x\in\{1,2,4\}}((x+1)+(x+3))=\frac c2\sum_{x\in\{1,2,4\}(2x+4)$

$1=\displaystyle\frac c2((2+4)+(4+4)+(8+4))$

$1=13c\implies\boxed{c=\dfrac1{13}}$