The joint probability function of random variables x and y...

The joint probability function of random variables X and Y is given by: y/x | 2 | 4 | 1 | 0.10 | 0.10 | 3 | 0.20 | 0.30 | 5 | 0.10 | 0.20 | (a) Give the marginal density of both X and Y as well as the probability distribution of X given Y = 3. (b) Give E(X) and Var(X). (c) Give E(X|Y=3) and Var(X|Y=3). (d) Give the covariance of X and Y.

Transcript

00:01 So we have the table for the joint probability function of random variables x and y given here.

00:09 And the first question is to get the marginal density of x and y.

00:13 So it is very easy to get the marginal densities.

00:16 We just need to calculate the row thumbs and column sums.

00:20 So for y equals 1, we have the probability point 2.

00:25 Y equals 3, we have 0 .5.

00:27 Y equals 5 we have probability 0 .3 and for x we have the two.

00:35 So for x the marginal density is f x is equal to 0 .4 for x being 2 and 0 .6 for x being 4 and 0 .6 for x being 4 and for y the density is given like for y equals let's make clear for y equals 1 and 0 .5 for y equals 3 and 0 .3 for y equals 5 for y equals 5.

01:17 So this is the marginal densities of x and y and then we can calculate the conditional probability of x given y being three so we just need to focus on this for y equals three the conditional probabilities so again we have two possibilities the first one is x equals 2 and the other one is x equals 4 so for this we just need to calculate point 2 over the probability of f y of y equals 3 which is 0 .5 and the second one is 0 .3 divided by 0 .5 so this is equal to 0 .4 this is equal to 0 .6 this is the marginal density of x given y equals 3 and for the second question b, we calculate the expectation and variance of x, because x just has two possibilities that x is equal to 2 times 0 .4 plus 4 times 0 .6, which is equal to 2 .3 .2.

03:04 And the variance of x is equal to 0 .4 times 2 minus the expectation is this one.

03:17 Or a better way to do this is that we can use the formula of variance of x is equal to the expectation of x square minus the square of expectation of x.

03:33 Right? so for x square, we know that it has a probability of 0 .4 for being 2 square, which is 4, plus probability of 0 .6 times 4 square, which is 16.

03:52 And then we have, it is 1 .6 plus 9 .6, which is equal to 11 .2.

04:04 And we need to minus the square of the expectation, which is 0 .2 square, minus 0 .2 square.

04:16 0 .2 square is equal to 10 .24.

04:22 And then we have this is equal to, let's just use the calculator to do that, which is 690.

04:38 9 6.

04:39 This is the variance of x and the variance of y are no, don't have to calculate that...