Answer to Question #104474 in Statistics and Probability for BARUN YADAV

Question

If X and Y are independent Poisson variates such that
P(X=1)=P(X=2) and P(Y=2)=P(Y=3).
Find the variance of X- 2Y.

Answer 1

Let X follow Poisson($\lambda$)

P(X=1) = $\frac{e^{-\lambda}\lambda^1}{1!}$

P(X=2) = $\frac{e^{-\lambda}\lambda^2}{2!}$

Using P(X =1)=P(X=2) we get $\fbox{$\lambda = 2$}$

Let Y follow Poisson($\lambda'$ )

P(Y=2) = $\frac{e^{-\lambda'}\lambda'^2}{2!}$

P(Y=3) = $\frac{e^{-\lambda'}\lambda'^3}{3!}​$

Using P(Y =2)=P(Y=3) we get $\fbox{$\lambda' = 3$}$

Since, X and Y are independent random variables,

Var(X-2Y) = Var(X) + 4Var(Y) = $\lambda+4\times\lambda'$

= 2 + 4$\times$3 = 14

This is because Variance of a poisson random variable is its parameter itself