Customers arrive at a store according to a Poisson process of rate $λ$ customers per hour. Each makes a purchase with probability $p$, independently. Given that a customer makes a purchase, the amount spent has mean $µ$ (in dollars) and variance $σ^2$.
(a) Find the mean and variance of how much a random customer spends (note that the customer may spend nothing).
Let number of custumers be $N$~Pois(λt); $P(I_p=1)=p$; let $X_n$ be the amounth spent. Then, $E[X_1|I_p=1]=µ$; $Var(X_1|I_p=1)=σ^2$
$E(X_n|N=n)=E(X_n|I_p=1,N=n)p+E(X_n|I_p=0,N=n)q=Nµp$
$E[X_n]=E[E(X_n|N=n)]=E[Nµp]=µpE[N]=λtµp$
$Var(X_n)=E[Var(X_n|N=n)]+Var[E(X_n|N=n)]$, where
$Var(X_n|N=n)=E[Var(X_n|I_p=1,N=n)]+Var[E(X_n|I_p=1,N=n)]=E[Nσ^2]+Var[Nµ]=λtσ^2+λtµ^2=>Var(X_n)=E(λtσ^2+λtµ^2)+Var(Nµp)=λt(σ^2+µ^2+µ^2p^2)$
Is it right/wrong?
Let $Y$ denote the money spend by a random customer under condition that this customer makes a purchase. Then $\mathbb EY=\mu$ and $\mathsf{Var}Y=\sigma^2$.
Let $B\sim Bernoulli(p)$ and let $B$ and $Y$ be independent.
Then you are asked to find the expectation and variance of $BY$.
Can you take it from here?