Suppose X is Binomial(3,1/2) and Y is Binomial(2,1/2). Let Z = 2X - 3Y. Assume X and Y are independent.

Question

Suppose X is Binomial(3,1/2) and Y is Binomial(2,1/2). Let Z = 2X - 3Y. Assume X and Y are independent. What is the smallest value of Z? What is the largest value of Z? I am unsure on how you would calculate these two values. Also, how would you computer P(Z=1) by hand to get an exact fraction? I have simulated this and got approximately 0.19.

Answer 1

One simple straightforward way is to write down all $12$ possibiliteis for $(X,Y)$. $-6$ is the smallest and $+6$ is the largest value. $Z=1$ is possible only when $Y=1$ and $X=2$ so $P(Z=1)=\binom {3}{2} (\frac 1 2)^{3} \binom {2} {1} ((\frac 1 2)^{2}$.