00:01
All right, so the first exercise, we're given a pdf, fx of x, cx, zero otherwise.
00:18
And the first part, we're asked to find c.
00:22
And so the way we do this is we integrate cx on the interval, on the support, zero to two.
00:32
And because it's a pdf, the total probability should be equal one.
00:36
So we integrate this and set equal to one, it's all for c.
00:39
So we get c x squared over two value is zero to two so we get four c over two which is two c which equals one and that tells us c is a half and then we want to find the probability that x is between zero and one now this the inequality signs are on the problem statement are given as less than or equal to and less as opposed to strictly less than and because it's a continuous distribution, it really kind of went in the same, doesn't matter because any particular value has a probability of zero.
01:20
If it's discrete, that's a whole story.
01:23
So what we do here is we integrate our distribution.
01:27
1 half x on the interval 0 to 1.
01:30
Integrate with respect to x.
01:32
So we get 1 half x squared over 2.
01:35
We value it on 0 to 1.
01:38
So we get 1 over a quarter.
01:41
The next problem asks us to find the cdf of x and that's the integral of the pdf with respect to x.
01:53
It valued on its support.
01:55
Well, actually no, in value between zero to x.
01:57
I think in textbooks it's listed as negative infinity up to x.
02:03
So for our purposes, we're going to change this a little bit.
02:08
So we've got a half x.
02:11
We want to go from zero to x, but you can see we're going to have a problem here.
02:15
Well, just some confusion because we have x.
02:17
On our interval, but here we have x on our, this is a variable that we're integrating with respect to.
02:24
So we just change this to a u.
02:30
All right, so then we integrate, and we get one half u squared over two, that we're zero to x, and we get x squared over four.
02:44
And there we go.
02:46
All right, now the next problem asks us to do some work with the z distribution.
02:52
And before we get into it, we'll just state the problems and we'll do that.
02:55
Show and do the calculations.
02:59
The first one asked to find the probability that z is between negative 1 and 1.
03:06
And then we're asked to find the probability of being between negative 2 and 2 in the z distribution.
03:14
And the same thing, but with 3 and negative 3.
03:21
And then the last, we want to find the probability of being greater than 3.
03:29
So how we do this is we take the probability of z being less than one minus the probability of z being less than negative one and same thing for negative two and two it's the probability of z being less than two minus the probability of z being less than negative two for three probability of z being less than three minus the probability of z being less than negative three and then this one probably z being greater than three will this notice we have probability of z being greater than three will this notice we have probability of z being less than three.
04:08
Defined greater than three, all we do is one minus the probability of z being less than three.
04:17
All right.
04:18
So i've done all the work already, and i've done all this work with a nice little spreadsheet function.
04:28
Let me tell you what it is.
04:30
I'll put it down here.
04:32
It's norm s.
04:35
Dist.
04:36
And i'm using google sheets.
04:38
Excel has a very similar formula.
04:41
So it's norm.
04:43
S dist and what you do is you put in your z value and then out pops the probability we want.
04:53
So it's just this easy for us to see.
05:11
So here, the probability of z being less than one using that function is this.
05:19
0 .84 and we probably, z being less than negative 1 is 0 .158.
05:24
So 0 .841 minus 0 .158 is 0 .62.
05:28
For two, we do the same thing with that lovely spreadsheet function right here.
05:37
We get .977.
05:40
.223, and we get 0 .954.
05:49
And then for three, i forgot a parentheses...