What is the distribution created by this?

Question

So I have a group of people. Each of them is 60% likely to vote on A and 40% likely to vote on B. What type of distribution does this create if I'm looking for amount of people that vote on A - the amount of people that vote on B? I'd say it would just be a difference of binomials, but they are dependent on each other so I'm not sure.

Answer 1

It's just a shifted, scaled Binomial. Essential, you want a weighted Binomial: $X_1,\dots,X_n$ are i.i.d. Bernoulli random variables with parameter $3/5$, and you are looking for $$ X = \sum_{k=1}^n (2X_k-1) = 2\sum_{k=1}^n X_k -n = 2Y-n $$ where $Y$ is a Binomial with parameters $n$ and $3/5$.

Answer 2

Let $$ I_{n}=\begin{cases} 1 & \text{if the }n\text{-th individual votes for }A\\ 0 & \text{otherwise}. \end{cases} $$ Assuming voters do not influence each other, then $I_{1},\ldots,I_{N}$ are i.i.d. $\operatorname{Bernoulli}(0.6)$ random variables. Let $V_A$ be the total number of voters for $A$ and $V_B$ be the total number of voters for $B$. Then, $$ V_{A}-V_{B}=\left[I_{1}+\cdots+I_{N}\right]-\left[N-\left(I_{1}+\cdots+I_{N}\right)\right]=2\left(I_{1}+\cdots+I_{N}\right)-N $$ where $I_{1}+\cdots+I_{N}$ is a binomially distributed $\operatorname{B}(N,0.6)$ random variable.