00:01
Hello students, as per the given question, let us solve the joint probability distribution of x and y.
00:13
So, in order to describe the joint probability distribution of x and y, we need to list all the possible outcomes of x and y where a fair die is rolled.
00:22
So, the possible outcomes of x are 1, 3, 4, 5 and 6.
00:32
These are the 6 possible outcomes when a fair die is rolled.
00:35
So, the x is odd, then y is equals to 1.
00:47
If x is even, then y is equals to 0.
00:53
So, let us draw a joint probability distribution that can be represented in the table below.
01:04
As we need to find the joint probability for this values where it is given that 1 for the odd numbers and 0 for the even numbers.
01:12
As 1 is the odd number, it is 1 and 2 is even.
01:15
So, it is 0, 1, 0 and 1 and 0 correspondingly.
01:22
And now, let us go and check for the independence of, we need to check for the independence where it gives that in order to determine x and y are independent or not, we need to check if the joint probability of x and y is equal to a product of their independent probabilities or not.
01:45
So, for that in order to check that we have probability of x is equals to 1 is equals to p of x is equals to 3 is equals to p of x is equals to 5.
01:56
As it is equals to 1 by 6, since the die is fair, each outcome has a possibility of a probability of 1 by 6.
02:06
And also the probability of x is equals to 2 is equals to probability of x is equals to 4 is equals to probability of x is equals to 6.
02:17
Even this has the 1 by 6 with the same reason having the each and every probability having the same equal number of occurrence as it is a fair die...