The joint p.m.f. of X and Y is f(x, y) = 1/6, 0 ≤ x + y ≤ 2, where x and y
are nonnegative integers.
(a) Sketch the support of X and Y.
(b) Record the marginal p.m.f.’s f1(x) and f2(y) in the “margins.”
(c)Compute Cov(X, Y).
(d)Determine ρ,the correlation coefficient.
(a) To sketch the support of X and Y, we need to identify the range of possible values for X and Y. From the given information, we know that X and Y are non-negative integers and x + y should be between 0 and 2.
Using this information, we can create a table showing all possible values of X and Y and shading the valid region.
| X\Y | 0 | 1 | 2 |
| --- | - | - | - |
| 0 | ✓ | ✓ | ✓ |
| 1 | ✓ | ✓ | ✗ |
| 2 | ✓ | ✗ | ✗ |
No values of Y can be greater than 2 because the sum of x and y should be between 0 and 2. From the table, we can see that X and Y can take values in the range [0, 2] and [0, 2] respectively.
(b) To find the marginal p.m.f.'s f1(x) and f2(y), we need to sum up the joint p.m.f. over the respective variables.
For f1(x), we sum up the joint probability over all possible values of Y for a fixed X:
f1(x) = Σ f(x,y) for all y
= f(x,0) + f(x,1) + f(x,2)
= 1/6 + 1/6 + 1/6
= 1/2
Similarly, for f2(y):
f2(y) = Σ f(x,y) for all x
= f(0,y) + f(1,y) + f(2,y)
= 1/6 + 1/6 + 1/6
= 1/2
So the marginal p.m.f.'s for X and Y are both 1/2.
(c) To compute Cov(X, Y), we use the formula:
Cov(X, Y) = E(XY) - E(X)E(Y)
The expected value E(XY) can be found by summing up the product of X and Y weighted by their joint probabilities:
E(XY) = Σ (x * y) * f(x,y) for all x and y
= 0 * 0 * (1/6) + 0 * 1 * (1/6) + 0 * 2 * (1/6) + 1 * 0 * (1/6) + 1 * 1 * (1/6) + 1 * 2 * (1/6) + 2 * 0 * (1/6) + 2 * 1 * (1/6)
= 1/6 + 2/6 + 2/6 + 2/6
= 7/6
The expected values E(X) and E(Y) are simply the sum of all possible values of X and Y weighted by their probabilities:
E(X) = Σ x * f1(x) for all x
= 0 * (1/2) + 1 * (1/2) + 2 * (1/2)
= 1
E(Y) = Σ y * f2(y) for all y
= 0 * (1/2) + 1 * (1/2) + 2 * (1/2)
= 1
Cov(X, Y) = E(XY) - E(X)E(Y)
= 7/6 - 1 * 1
= 1/6
(d) To determine the correlation coefficient ρ, we need to find the standard deviations of X and Y.
The variance of X can be found using the formula var(X) = E(X^2) - (E(X))^2.
E(X^2) = Σ x^2 * f1(x) for all x
= 0^2 * (1/2) + 1^2 * (1/2) + 2^2 * (1/2)
= 2
var(X) = E(X^2) - (E(X))^2
= 2 - 1^2
= 1
Similarly, the variance of Y is also 1.
Now we can calculate the correlation coefficient:
ρ = Cov(X, Y) / (√(var(X)) * √(var(Y)))
= (1/6) / (√1 * √1)
= 1/6
So the correlation coefficient ρ is 1/6.