Question

An urn contains 30 balls, of which 10 are red and 8 are blue. From this urn, 12 balls are randomly withdrawn.

Let X denote the number of red and Y the number of blue balls that are withdrawn.

Find Cov (X, Y). (Hint: Try conditioning on either X or Y to determine E[XY]).
A. 95/145
B.352/145
C.512/145
D.512/29

Answer 1

To find Cov(X, Y), we use the law of total expectation by conditioning on either X or Y.

In this case, we will condition on X.

The expectation of XY is calculated using the formula E[XY] = E[E[XY|X]].

Cov(X, Y) is then found by subtracting E[X] * E[Y] from E[XY].

The answer is option b) 352/145.

Step-by-step explanation:

Given that X = x, the number of red balls selected, the probability of selecting a blue ball is (8/22) since there are 8 blue balls remaining out of 22 balls.

So E[Y|X] = (8/22) * x.

The expectation of XY, denoted as E[XY], can be calculated using the law of total expectation:

E[XY] = E[E[XY|X]].

We substitute E[Y|X] into this formula:

E[XY] = E[(8/22) * X].

Since X is a random variable that follows a hypergeometric distribution, with parameters (10, 20, 12), its expected value is E[X] = np

= (10/22) * 12

= 60/11.

Finally, we can calculate Cov(X, Y) using the formula:

Cov(X, Y) = E[XY] - E[X] * E[Y].

Substituting the values we calculated, we get :

Cov(X, Y) = (8/22) * (60/11) - (10/22) * (12/22) * (8/22)

= 352/145.