find value of c for the Probability mass function

Question

p(x, y) = c(x + y), x = 1, 2, 3, y = 1, . . . , x

I tried solving it by integrating it over y and them summing it up over x. But I have a doubt can I integrate the function since it is a probability mass function. Please help.

Answer 1

You need to compute $\sum_{x=1}^3\sum_{y=1}^x (x+y)$ and seem to be having trouble with the inner sum. It splits up into two terms. The first is easy cause the $x$ just comes out: $$ \sum_{y=1}^x x = x\sum_{y=1}^x 1 = x^2.$$ For the other we have the formula $$ \sum_{y=1}^x y = \frac{x(x+1)}{2}.$$

Then you need to sum the resulting expression from $x=1$ to $3,$ which can be done by either looking up / figuring out the formulae for $\sum_{i=1}^n i^2$ or more simply, by hand since it's only three terms.

Answer 2

Note that $$\sum_{y=1}^x c(x+y) = cx^2 + c\frac{x(x+1)}{2}~.$$ Now, sum this function of $x$ from $1$ to $3$ and equate the resulting expression to $1$ to get $c$.

Answer 3

Evaluate $c$ when the total probability mass equals 1.$$1{~=~\sum_{x=1}^3\sum_{y=1}^x c(x+y)\\~=~ c\sum_{x=1}^3(x^2+\sum_{y=1}^x y)\\~=~c\sum_{x=1}^3(x^2+x(x+1)/2)\\~=~ \frac c2\sum_{x=1}^3(3x^2+x)\\ ~~\vdots}$$