00:01
All right, so we have a uniform distribution that runs between negative 4 and 6, and we want to use integration to calculate the expected value of the variance.
00:09
All right, so this is a fun thing.
00:11
So first we need our uniform distribution, get our function.
00:16
Well, that equals 1 over, 1 over, well, b minus a, we'll say.
00:29
In this case, a and b are here, so this is our starting values, or ending value, 1 over.
00:35
This is going to be 6 minus negative 4, so it's going to be 1 tenth.
00:40
So this is our uniform distribution right here.
00:44
So the expected value is given as x times f of x integrated with respect to x.
00:56
So we just, oh, and evaluated on our interval.
01:03
So we do that here.
01:04
So let's see what we got.
01:05
We have, it's going to be an integral from negative 4 to 6 of, x times one -tenth d x well the one -tenth we can bring out front it's a constant now we just integrate this thing so we get a tenth times let's see it's going to be x squared over two evaluated between negative four and six so let's see what we got we have one -tenth we're going to have let's see it's six squared over two minus x square 4 squared over 2.
01:57
Let's see what else we're going to get.
01:59
We're going to get, this is a 10th.
02:04
And here we have, what's that, 36 minus 16, so 20 has, right, because that's 36.
02:15
And this is going to be 16 or subtracting them.
02:18
So it's going to be 20.
02:19
Yep, there we go.
02:20
So then the expected value is equal to, let's see, 20 over 2 is 10.
02:29
So, oh, it's going to be 1.
02:31
Yeah, so it's going to be 1.
02:31
It's kind of interesting.
02:34
It's fun.
02:37
Now we're going to do the variance of x, and that's equal to the expected value of x squared minus the expected value squared.
02:51
So it's good.
02:51
We know this part already.
02:54
We know, we can do this over here.
02:57
We're going to do a little final calculation over here.
03:00
We don't know this yet, but we do know expected value is 1, so 1 squared is just 1.
03:06
But we need this piece.
03:08
So let's do that.
03:09
X squared.
03:11
This is a separate.
03:17
Let me read this over here.
03:21
The expected value of x squared is equal to, well, x squared times f of x.
03:29
And we know f of x.
03:30
Oh, integrate with respect to x, of course.
03:32
Well, this is just going to be one -tenth.
03:37
The value of the integral of negative is 4 to 6.
03:41
So let's see.
03:42
Let's churn through this.
03:44
Let's simplify that...