Let the discrete random variable X have probability mass function $p(x)=c*2^{x}*x!^{-1}$ for x = 0,1,... and zero otherwise. What is the constant c?

Question

I know that the sum of p(x) for all x should be 1, but as it's countably infinite, I'm having trouble figuring out the solution. I've tried taking the limit of p(x) as x -> infinity, but I get zero using the ratio test (hopefully it works if I only technically sub largely infinite elements of N instead of R).

Answer 1

You correctly know that the sum of probabilities needs to be $1$:

$$\sum_{x=0}^{\infty}\mathbb{P}(X=x)= c\sum_{x=0}^{\infty}\frac{2^x}{x!} = 1$$

Do you know what the value of that sum could be? Once you know that, you can find $c$.

HINT: What is the series expansion of $e^x$?