Let X be a Poisson random variable with µ = EX = 0.4 and let Y be another

random such that
E[(2Y + 1)2
] = 10 and E[(Y − 1)2
] = 4
Consider the random variable Z such that Z = 3X + 4Y + 2.
1. Find E(X2
), E(Y
2
) and E(Z).
2. Suppose that X and Y are independent, find the variance of Z

To find E(X^2), you can use the formula for the variance of a Poisson random variable, which is Var(X) = E(X^2) - (E(X))^2. Since the mean µ of X is given as 0.4, we can use this formula to find E(X^2).

1. Finding E(X^2):
Var(X) = E(X^2) - (E(X))^2
0.4 = E(X^2) - (0.4)^2
0.4 = E(X^2) - 0.16
E(X^2) = 0.4 + 0.16
E(X^2) = 0.56

2. Finding E(Y^2):
E[(2Y + 1)^2] = 10
Expanding the square:
4E(Y^2) + 4E(Y) + 1 = 10
4E(Y^2) + 4E(Y) = 9
E[(Y - 1)^2] = 4
Expanding the square:
E(Y^2) - 2E(Y) + 1 = 4
E(Y^2) - 2E(Y) = 3

Multiplying the second equation by 2:
2E(Y^2) - 4E(Y) = 6

Adding the two equations together:
2E(Y^2) - 4E(Y) + E(Y^2) - 2E(Y) = 9 + 6 - 3
3E(Y^2) - 6E(Y) = 12
E(Y^2) - 2E(Y) = 4

Now, we have a system of equations:
E(Y^2) - 2E(Y) = 4
E(Y^2) - 2E(Y) = 4

Solving this system of equations, we find:
E(Y^2) = 4
E(Y) = 2

3. Finding E(Z):
Z = 3X + 4Y + 2
E(Z) = E(3X + 4Y + 2)
E(Z) = 3E(X) + 4E(Y) + 2
E(Z) = 3(0.4) + 4(2) + 2
E(Z) = 1.2 + 8 + 2
E(Z) = 11.2

Therefore, E(X^2) = 0.56, E(Y^2) = 4, and E(Z) = 11.2.

4. If X and Y are independent, the variance of Z can be found using the properties of variance. Since Var(aX + bY) = a^2Var(X) + b^2Var(Y) for independent random variables:
Var(Z) = 3^2Var(X) + 4^2Var(Y)
Var(Z) = 9(0.4) + 16(4)
Var(Z) = 3.6 + 64
Var(Z) = 67.6

Therefore, the variance of Z is 67.6 if X and Y are independent.