Verifying that binomial distribution is a valid PMF.

Question

I'm trying to figure out if X ~ Bin(n,p) is a valid PMF. In other words, I need to show that:

$$\sum_{x \in X} {n\choose x} p^{x} (1-p)^{n-x} = 1$$

Answer 1

You can use the binomial theorem to show that:

$$\sum_{x}\binom{n}{x}p^x(1-p)^{n-x} = [p + (1-p)]^n= 1^n = 1$$