X = The number of customers entering a shop. X∼Poisson(4)
P(Costumer buys 1 pair of socks) = 3/4.
decisions are not dependent.
Y random variable that equals the number of socks sold.
What is E[Y]?
Would like to get some help with this question, I got a hint that its between [3,3.5) but not sure how to get the exact value.. Thanks in advance.
We have $$P(X=n)=\frac{4^ne^{-4}}{n!}$$ The expected number of socks bought if $n$ customers enter is $\frac34n$. Then $$E(Y)=\sum_{n=0}^\infty\frac{4^ne^{-4}}{n!}\cdot\frac34n$$ $$=3\sum_{n=1}^\infty\frac{4^{n-1}e^{-4}}{(n-1)!}$$ $$=3\sum_{n=0}^\infty\frac{4^ne^{-4}}{n!}=3$$ The following more general property holds: if $X\sim\operatorname{Pois}(\lambda)$ and $Y$ given $X=k$ follows $\operatorname{binom}(k,p)$, then $Y\sim\operatorname{Pois}(p\lambda)$.