00:01
All right, so for this problem, we're given these conditional probability functions of y conditioned on x, and we're asked to calculate the following values.
00:10
So first, we need the expected value of y conditioned upon x equaling zero.
00:16
Well, we already have the conditional probability function of y conditioned on x equals zero.
00:21
And so we're just going to take the sum of the values of y of y times the probability of y given that x equals zero.
00:31
Which is equal to 0 times 0, which will just be 0, plus 1 times 0, plus 2 times 0, plus 2 times 0, which equals 1 .3.
00:48
Then the expected value of y conditioned on x equaling 1, we're going to calculate the same way.
00:54
It's going to be 0 times 0 .4 plus 1 times 0 .2, plus 2 times 0 .4, which equals 1, 2, 2 times 0 .4, which equals 1, 2 times 0 .2, 2 times 0 .4, 2 times 0 .2, 2 times 0, which point four is point eight plus point two is equal to one.
01:10
Now the expected value of x, we're told that x is either zero or one, and the probability of x is equal, of x equaling zero is point four.
01:18
So in other words, you can say you have x, p of x, 0 ,0, 1, 0 .4, then this has to be 0 .6.
01:28
So the expected value of x is just the sum of x times p of x, which is just going to be 1 times 0 .6 or 0 .6.
01:36
And the variance of x is equal to the expected value of x squared minus the expected value of x squared which in this case is still going to be the expected value of x squared it's going to be one or where is it one squared times 0 .6 which is still 0 .6 minus 0 .6 squared is 0 .36 and you get a variance of x equal to 0 .24.
02:10
All right.
02:11
Now we're asked to find the joint probability function of x and y.
02:17
And so why don't we write it as a table where we have x and y, y, x equals 0 or 1, y equals 0, 1 or 2? here we can actually make that a bit neater.
02:32
And i'm going to switch to make it in this direction.
02:41
So let's have, okay.
02:58
So we'll have x equals 0 and 1, y equals 1, or 0, or 0.
03:04
1 and 2.
03:06
So the joint probability, let's just use events a and b and use the conditional probability formula.
03:18
P of a given b is equal to the probability of a and b divided by the probability of b.
03:26
So similarly, we can say the probability of x equals a number and y equals a number is equal to, now you can take the conditional probability, the conditional probability that y equals a number given that x equals a certain number divided by the probability, or sorry, times the probability that x equals that number, that's equivalent to saying the probability of a and b is equal to p of a given b times b.
03:59
So we can take your conditional probabilities, probability of y equaling zero given that x equals 0 .2, and multiply by the probability of that x equals 0 of 0 .4, to get 0 .08.
04:16
And we can keep doing this.
04:17
0 .3 times 0 .4 is 0 .12.
04:21
And 0 .5 times 0 .4 is 0 .20.
04:25
And then 0 .4 times 0 .6, which is a probability that x equals 1 is 0 .24.
04:33
0 .2 times 0 .6 is 0 .12.
04:37
Finally, 0 .4 times 0 .6 is 0 .24.
04:41
And we can verify that this is a valid joint probability mass function of adding these individual values...
