Sum of two binomial variables

Question

I'm quite not sure how can I show that a sum of two binomial variables with different probability for success is NOT binomial. Can someone help me with this issue?

Thanks in advance

Answer 1

Assuming they are independent, if they were distributed $\text{Binomial}(n_1, p_1)$ and $\text{Binomial}(n_2, p_2)$ and the sum were $\text{Binomial}(n_3, p_3)$ you would need

• $n_3=n_1+n_2$ to match the support (assuming $p_1>0$ and $p_2 \gt 0$)
• $n_3p_3 = n_1p_1+n_2p_2 $ to match the mean
• $n_3p_3(1-p_3) = n_1p_1(1-p_1)+n_2p_2(1-p_2)$ to match the variance

This gives three equations in the two unknowns $n_3$ and $p_3$, which would give inconsistent results unless $p_1=p_2$ or $n_1=0$ or $n_2=0$