00:02
Hello, we are given that x and y are independent random variables and we have to prove the statement variance of xy is equal to expected value of x square times variance of y plus variance of x times expected value of y square plus variance of x times variance of y and we should use the law of total variance to prove this statement.
00:30
The law of total variance is variance of y is equal to expected value of variance of y given x plus variance of expected value of y given x.
00:51
Note that variance for a general pair of random variables x, y variance of y given x is a random variable.
01:00
It will depend on the value of x.
01:02
Similarly, expected value of y given x is a random variable.
01:08
It will depend on the value of x and we'll come to this.
01:14
Okay, so applying this law of total variance to x times y, we have variance of xy is equal to expected value of variance of xy given x plus variance of expected value of xy given x.
01:34
So, variance of xy is equal to but note that when x is given in xy, x is like a constant so that this thing is like variance.
01:47
So given so that this in this random variable given x, xy is like a constant times y so that variance of xy given x is like a constant times is like variance of a constant times y.
02:12
So, this variance is equal to x square times the x is like the constant x is a constant here.
02:23
Well, this thing is equal to x square times variance of y given x because we know that variance is a common, it's a well -known property that variance of a constant times a random variable is equal to c square times variance of x where c is a constant.
02:46
Similarly, expected value of c times x is equal to c times expected value of x where c is a constant.
02:53
Now, given x in xy, x is like a constant.
02:59
So variance of xy given x is like variance of a constant times y given x.
03:10
So this the variance of this thing is equal to x square times variance of y given x.
03:18
Similarly in here, given x in xy, x is like a constant.
03:24
So it is expected value of a constant times y given x.
03:29
So it's like x times...