Need Help finding constant C in joint pmf for two discrete variables

Question

Determine the constant c so that f (x, y) satisfies the conditions of being a joint pmf for two discrete random variables X and Y:

 

f (x, y) = c(1/4)^x * (1/3)^y, where x = 1, 2, . . . , y = 1, 2, . . . .

 

The answer is 6, but I need help reaching this answer. Any help would be appreciated.

Answer 1

The sum of a pmf over its entire domain must equal 1.

So cΣ(x=1 to ∞)Σ(y=1 to ∞)[(1/4^x)(1/3^y)] = 1.

 

Σ(i=1 to ∞)[1/n^i] = 1/(n-1). So

 

Σ(y=1 to ∞)[(1/3^y)] = 1/(3-1) = 1/2 and

Σ(x=1 to ∞)[(1/4^x)] = 1/(4-1) =1/3.

 

So, Σ(x=1 to ∞)Σ(y=1 to ∞)[(1/4^x)(1/3^y)] = (1/2)*(1/3) = 1/6.

 

To make the sum = 1, then c=6.