00:01
Hello everyone, so in this question, the joint distribution of x and y is given as shown in the table.
00:09
So we have to find first the marginal distributions of x and y.
00:14
So first we can convert this table in the form, y is equals to 0, y is equals to 1, x is equals to 0, x equals to 1 and x is equals to 2.
00:27
Let us write the values 1 divided by 8 1 divided by 8 1 divided by 4 1 divided by 8 1 divided by 4 and 1 divided by 8 here comes a marginal total total and here also marginal total total right so this becomes 1 divided by 8 plus 1 divided by 8 is 2 divided by 8 similarly 3 divided by 8 similarly 3 divided by 8, 5 divided by 8 and 3 divided by 8.
01:10
In total we have 1.
01:13
So here in part a we have to find the marginal distribution of x and y.
01:20
So marginal distribution of x is nothing but this part of the table whereas marginal distribution of y is this part of the table.
01:30
Writing it particularly in the form of table, part a is marginal distributions so first is of x which becomes value of x and probability of x is equal to x value of x is equal to small x right so we have x is equal to 0 1 2 and probability respectively being 2 divided by 8 3 divided by 8 3 divided by 8 and total being so this is for x and second marginal distribution of y becomes value of y, capital y is equal to small y and probability of capital y is equals to small y.
02:37
We get 0 and 1 probability being 5 divided by 8 and 3 divided by 8 total represented by.
02:50
So this is a part 8.
02:52
In part b, we have to find the conditional distribution of x b.
02:59
Given by conditional distribution of x given y is equals to 0.
03:13
So we have value of x is equals to small x and probability of capital x is equal to small x given capital y is equal to 0.
03:27
Now simply substituting the values, the values of possible x are 0 1 and 2.
03:32
The probability of x is equal to small x from the table capital x is equal to 0.
03:42
At y is equals to 0 is 1 divided by 8, whole divided by probability of y is equal to 0, that is 5 divided by 8 using this.
03:54
So we have 1 divided by 8, whole divided by 5 divided by 8, which becomes 1 divided by 5.
04:02
Similarly, we have 1 divided by 4 divided by 5 divided by 8 is equal to 2 divided by 5 and 1 divided by 4 divided by 5 is equal to 2 divided by 5.
04:16
Rating total total is equal to 1.
04:20
Again, conditional distribution of y given x is equal to 1.
04:34
So we will write value of capital y is equal to small y.
04:39
And the probability of capital y is equal to small y given x is equal to 1.
04:45
Possible values of capital y are 0 and 1.
04:48
Now from the table, if capital y is equal to 0 and x is equal to 1, then the value is 1 divided by 4.
04:58
So here it comes 1 divided by 4.
05:01
Now whole divided by probability at x is equals to 1.
05:05
So from this table probability that x is equal to 1 is 3 divided by 8.
05:09
So we have 3 divided by 8...