00:01
Alright, so we have three variables and we're told that x is uniform and we'll define that on 0 to 1.
00:08
And it's defined on a to b, but usually, not usually, but often times it's 0 to 1.
00:15
And then we've got variable y, which we're told is exponential, and that's got parameter, say, lambda.
00:23
You can see this data as well.
00:25
And then we've got variable z, which we're told is normal.
00:31
And that has a mean mu and a standard deviation sigma or variance sigma squared.
00:38
Or and variance sigma squared.
00:40
So let's talk about this.
00:41
And the question is, which of these three has the highest probability on the interval 2 to 4? so which of them, so probability on the interval 2 to 4, which has the highest probability.
00:57
So we're going to look at the distributions and let's look at the functions here.
01:00
So x, the function for the pdf probability density function is p of x is equal to, well, 1 if it's on a to b.
01:13
And this would be on the interval 0 to 1.
01:18
The exponential distribution, probability of y is going to be e to the negative x or i should say lambda.
01:31
E to the negative lambda.
01:32
And this is for x greater than 0.
01:37
Oops, i forgot the x here.
01:39
Sorry, y.
01:40
Y, y, y.
01:40
Pardon me.
01:41
Y.
01:42
And then lastly, z is uniform and it's got a fun function here.
01:46
P probability of z is 1 over root 2 pi times sigma outside the square root.
01:53
E to the negative x minus mu squared over sigma or 2 sigma.
02:01
All right.
02:02
So which of them has the highest probability on 2 to 4? let's look at that.
02:09
So first off, x, the shape of x will be a rectangle on 0 to 1.
02:16
So it's going to be a square like this.
02:20
Here's 0.
02:21
Here's 1.
02:22
Like that.
02:23
So it should be contained there.
02:25
Y is going to look like this.
02:29
And this is the exponential distribution.
02:32
Look like this and it's skewed down like that.
02:36
And then z, which is normal.
02:47
So for us, for which of them is going to have the highest probability here on 2 to 4? well, x doesn't contain 2 to 4 as it's defined here.
02:58
If we define it for a as 0 and b as 1.
03:02
It's not even there.
03:04
So it doesn't exist.
03:06
Y, this goes on for ever and ever and ever.
03:11
So 2 to 4, we'll say is right here.
03:15
The area we're looking for is this area.
03:16
This is the probability here.
03:18
This is the probability of y being between 2 and 4 inclusive.
03:31
Whereas here, the normal distribution, if we talk about the mean mu variance of 1, the center of our distribution would be 0 and 2 would be out here, 4 is up here.
03:43
And we want this area.
03:45
And this is the area we want.
03:46
This is the probability that z is between 2 and 4.
03:52
The question is which of those areas is bigger? and x will not touch it at all.
03:56
So it won't be x.
04:00
And let's look at it...