1. The following table gives the joint probability function...

1. The following table gives the joint probability function for random variables X and Y: Values of X Values of Y | 0 1 2 0 | 0.3 0.2 0.1 1 | 0.2 0.1 0 2 | 0.1 0 0 a) Compute P(X = 1, Y = 1) b) Compute P(X = 1 | Y = 1) c) Compute F_{X,Y}(0,2) d) Find the marginal probability function of random variable X, P(X = x) e) Compute Cov(X,Y)

Transcript

0:00 Hello, let's have a look on the question.

00:01 So this is the hello, let's have a look on the question.

00:04 So this is the table given to us.

00:06 We have joint probability function for x and y.

00:09 For the a part, we need to find out the probability of x is equal to 1 and y is equal to 1.

00:14 So in this case, if we check the table, when x is equal to 1 and y is equal to 1, the common value is 0 .1.

00:22 This is the probability which is 0 .1.

00:24 Moving on to the b part, we need to find out the probability of x is equal to 1 given y is equal to 1.

00:33 So in this case, this will be equal to probability of x is equal to 1, y is equal to 1 divided by probability of 1.

00:44 So this will be equal to, we already found out this value in the part 1, that is 0 .1 and the probability of 1 will be adding all the three probabilities, these three probabilities, which is 0 .2 plus 0 .1 plus 0, which is equal to 0 .1 divided by 0 .3, which is equal to 0 .333.

01:11 Moving on to the c part, here we need to find out the f of x, y for 0 and 2.

01:23 This will be equal to, this can be written as probability of x is less than equal to 0 and y is less than equal to 2.

01:30 Now here, x has three values but above condition only satisfies 0 and similarly y has three value and it satisfies all values.

01:38 So we can write that this probability x is less than equal to 0 and y is less than equal to 2 will be equal to probability of x is equal to 0, y is equal to 0 plus probability x is equal to 0, y is equal to 1 plus probability x is equal to 0 and y is equal to 2 because it is taking all three values.

02:00 This will be equal to 0 .3 plus 0 .2 plus 0 .1, which is equal to 0 .6.

02:07 So we can write f of x, y for 0 and 2 will be equal to 0 .6.

02:16 Next we have for the d part.

02:18 We need to find out the marginal probability function of x.

02:24 Marginal probability function will be for x is equal to 0.

02:28 We will find out the probability of x at its every point.

02:32 So we will add all these probabilities.

02:34 So for x equal to 0, we have 0 .3 plus 0 .2 plus 0 .1, which is equal to 0 .6...

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