00:02
We are given a random variable y.
00:07
Y is a random variable.
00:10
We know its mean value.
00:12
If we write it in the expectation form, we know the expectation of y is equal to 1, and the variance of y is equal to 4.
00:21
Now we define a new random variable by y.
00:26
I mean z is equal to 1 half times y minus 1.
00:29
We want to compute the expectation of z, which is also called the mean value of z, and also we want to compute the variance of z.
00:41
Now by the definition, the expectation of z will be just equal to expectation of 1 half times y minus z.
00:50
We know the mean value or the expectation is linear.
00:53
That means we can plot 1 half at the very first beginning.
00:57
It will be equal to 1 half times the expectation of y minus 1.
01:04
Again, the expectation is linear, so we know it will be equal to 1 half times expectation of y minus 1 half times the expectation of 1.
01:16
This guy, the expectation of 1 is just 1.
01:22
That means if we have a random variable which is a constant 1, then the mean value of this constant 1 is again 1.
01:33
It will be 1 half times the expectation of 1 minus 1 half.
01:38
By our assumption, we know that expectation of y is just 1, so we know it will be 1 half minus 1 half, which is 0, just as required, as hinted in the question.
01:57
We prove the mean value or the expectation of z is 0.
02:02
Now we want to consider the variance of z.
02:09
By the definition, the variance will be the expectation of z minus the expectation of z to the power 2.
02:18
This is a way to compute the variance...