Question

Let X represent the number of customers arriving during the morning hours and let Y represent the number of customers arriving during the afternoon hours at a diner. You are given

1. X and Y are Poisson distributed
2. The first moment of X is less than the first moment of Y by 8
3. The second moment of X is 60% of the second moment of Y

Calculate the mean and variance of X and Y

Answer 1

here as first moment is E(X) and E(Y)

therefore E(X)+8 =E(Y) .............(1)

also E(X²) =0.6*E(Y²).................(2)

as for poisson distribution E(X²) =E(X)+(E(X))²

therefore E(X)+(E(X))² =0.6*(E(Y)+(E(Y))² )

E(Y)-8+(E(Y)-8)² =0.6*E(Y)+0.6*(E(Y))²

-15.6*E(Y)+56+0.4(E(Y))² =0   

solving above E(Y) =35

and E(X) =27

therefore mean of X =27

and Variance of X =27

also

mean of Y =35

and Variance of Y =35