# Binomial Random Variable Multiplied by a constant

I have a Binomial(n,p) random variable that is multiplied by a constant, specifically 1/n. I know for some random variables multiplying by a constant doesn't change the distribution, i.e. the Normal, but for others it does, i.e. the Poisson. I can't find anything about the Binomial when it is multiplied by a constant. I know the expected value and variance but what I'm really interested in is whether this is a valid (i.e. not improper) random variable for all constants or just for some. I would appreciate any guidance.

Let us denote the constant by $c$ and the random variate itself by $x$. The original, binomial, random variate is multiplied by $c$ to get $x$. The probability distribution of $x$ is:
$$p(x;n,p,c) ={n \choose x/c}p^{x/c}(1-p)^{n-x/c}$$
except for $c=0$ of course. This works because $x/c$ transforms $x$ back to the original Binomial variate, which, in the case of discrete random variables, is all you need.
• The support would be $\{0, 1/n, 2/n, \dots, 1\}$ if $c = 1/n$. You don't have to have an integer value for a discrete random variable, just a countable number of possible values; whether they are integer values or not is irrelevant. Commented Jun 15, 2018 at 19:01