00:01
In this question, we're given the definition of variance of x.
00:03
It will be expectation of x square minus expectation of x squared.
00:09
Now we also have expectation of, for example, ax plus b.
00:14
It will be a times expectation of x plus b.
00:21
Now with this, we are able to do part a now.
00:26
That is, we want to show that variance of x plus b is equal to.
00:34
Variance of x so let's start with the left hand side now variance of x plus b so let's apply this definition will be expectation of the entire x plus b square minus expectation of x plus b the whole thing squared now let's expand out inside here we have x square plus bx plus b square minus now here here here expectation of x plus b so within the bracket here will be expectation of x plus b when you expand up using this okay now from here this can be also expanded out into expectation x square plus expectation of bx plus expectation of b square which is just going to be b square because it's a constant minus over here, let's expand out the square.
01:46
It will be expectation of x, the whole thing square, plus b times expectation of x plus b square.
01:56
Okay, so this is expectation of x square.
02:00
Now this b can be brought out.
02:02
It will be b, ex, can be brought up because of this.
02:07
See, ax, it can be brought out like this, plus b square minus expectation of x.
02:15
The whole thing square.
02:16
Let's expand up, move the minus sign in, minus b square.
02:23
Now you can see that some stuff can be cancelled.
02:26
This and this are, it's cancelled.
02:28
This and this is cancelled.
02:30
So we are left with ex square minus ex the whole thing square.
02:38
And this is exactly variance x...