Question

The random variables, X and Y , have the joint pmf f(x,y)=c(x+2y), x=1,2 y=1,2 and zero otherwise.

1. Find the constant, c, such that f(x,y) is a valid pmf.

2. Find the marginal distributions for X and Y .

3. Find the marginal means for both random variables.

4. Find the marginal variances for both random variables.

5. Find the correlation of X and Y .

6. Are the two variables independent? Justify.

Answer 1

1) f(x, y) = c(x + 2y)

P(X = 1, Y = 1) = 3c,
P(X = 2, Y = 1) = 4c,
P(X = 1, Y = 2) = 5c,
P(X = 2, Y = 2) = 6c

Sum of all probabilities should be 1. Therefore 18c = 1

Therefore c = 1/18

2) The marginal distributions for X and Y are obtained here as:

P(X = 1) = 4/9 and P(X = 2) = 5/9
P(Y = 1) = 7/18 and P(Y = 2) = 11/18

3) The marginal means are computed here as:

E(X) = 1(4/9) + 2(5/9) = 14/9
E(Y) = 1(7/18) + 2(11/18) = 29/18

4) The marginal variances are computed as:

E(X²) = 1(4/9) + 2²(5/9) = 8/3
E(Y²) = 1(7/18) + 2²(11/18) = 17/6

Var(X) = E(X²) - [E(X)]² = (8/3) - (14/9)² = 20/81
Var(Y) = E(Y²) - [E(Y)]² = (17/6) - (29/18)² = 77/324

5) E(XY) = 3c + 8c + 10c + 24c = 45c = 45/18 = 15/6 = 5/2

Cov(X, Y) = E(XY) - E(X)E(Y) = (5/2) - (14/9)*(29/18) = -1/162

Therefore correlation here is computed as:

6) As cov(X,Y) is not equal to 0, therefore the 2 variables are not independent here.