00:01
So in this question, we are given a joint pdf that's c times 1 minus y, where 0 is less than x is less than y is less than 10.
00:17
Sorry, no, less than this last one.
00:19
This is confusing to see.
00:20
It's 0 is less than x is less than y is less than 1, and it's 0 otherwise.
00:25
And we're asked to find the value of c that makes this a valid joint pdf.
00:30
So just to give us a good quick little graph of what this looks like, this is x, this is y, this is 1, this is 1.
00:40
And we know here that x goes between 0 and 1, and for any value of x, let's say like a half, y has to be greater than it.
00:50
So you can kind of do a little bit of playing around, and it looks like this is the region that we care about.
00:59
That is the region we're taking the pdf on.
01:01
So we're going to just go and do some integrals to compute this stuff.
01:05
First thing you can do is you can say, you want to integrate over this region.
01:09
We'll just integrate in the middle from x, on the outside, from x from 0 to 1.
01:16
So i'll just do that.
01:18
And then on the inside, we'll integrate from x to 1y.
01:22
And we want to integrate our joint pdf.
01:25
We expect this to integrate to 1.
01:28
So if i just go and just do this by hand, or you can plug it into a calculator, just for convenience, i will just plug this into a calculator so that we can not waste a ton of time here.
01:46
So this inner one i see calculates out to 1 half c times minus 1 plus x quantity squared.
02:02
And then when i integrate that from 0 to 1, i get c over 6.
02:14
So i get c over 6.
02:15
And just to say it on space, i'll say, i get c over 6 equals 1, or c equals 6.
02:21
So we conclude that we should take c equal to 6 to make this distribution reasonable.
02:28
Then the next piece is to find, maybe we'll make sure we don't go super long here.
02:35
We'll just talk about probability.
02:36
Probability x is less than 1 half when y is greater than 2 thirds.
02:45
So what's probability x? so the wording here is implying that we want a given, or a conditional probability.
02:53
We want the probability that x is less than 1 half given y is greater than 2 thirds.
02:57
Which, recall, that's the probability that both these are true, given just over just the probability that y is greater than 2 thirds.
03:14
Both of these are integrals.
03:16
So maybe i'll let you go and work through the integrals.
03:24
But this shouldn't be too difficult.
03:27
We just want to integrate.
03:29
Let's see.
03:31
What's the easiest way to do this? so on the first one, we can just integrate 0 to 1 half for x.
03:48
We can do a similar integral.
03:50
And then this one is going to be x is always less than 2 thirds.
03:55
So we'll want to integrate that.
03:56
And then it'll be our joint pdf, equal to idx.
04:02
So very, very similar to before.
04:07
Let's see what we get when we do this.
04:10
So when i do this, i get 1 over 216 for the top.
04:30
So we still have to do the bottom.
04:33
But i get that the top is 1 over 216.
04:38
And the bottom, we have to integrate y greater than 2 thirds...