00:01
Let us plot this region carefully because this is a region given.
00:06
Let's plot it carefully.
00:09
This is 0 -1 -2 -0 -1 -2.
00:16
And y is equal to x -strait line will be this.
00:20
This is y -equal to x -s straight line.
00:23
And x is equal to 2 minus -y straight line means x -plus y is equal to 2.
00:27
That looks like this.
00:28
This is x -pice -y is good -shut.
00:33
And this is 1.
00:36
So basically this is 1 come -over.
00:39
1 and what is the region x is less than 2 minus y x is less than 2 minus y means x plus y less than 2 so it should be below the green color line so this is x plus y is equal 2 so it's below the line now if you can see y is taking from 0 to 1 only 0 to 1 so that means y is taking this part so why is taking this part and y is less than x means x is greater than y okay x is greater than y means it is below this line below.
01:20
I'm sorry i told above it is below.
01:22
So, x is greater than y is below.
01:25
So that means this is the region of integration, this one, this triangle.
01:32
This is the region.
01:33
So this is the region where this pdf is defined, it's triangular region.
01:38
And this is the coordinates 1 comma 1.
01:39
This is origin.
01:41
This is 2 .0.
01:43
All right.
01:44
So now let's start finding the marginal density functions.
01:48
Fx of x will be the support of y, the joint pdf, d .y, the support of y, the joint pdf, f of x comma y, dy.
02:07
So what is the support of y? that is the question now.
02:11
So this is a very important task.
02:15
So what we do is we'll break up this triangular region into two regions.
02:19
So let me break up this into two regions.
02:21
So that means i'll draw a perpendicular, this region, say r1, this region r2.
02:31
So that what we do is let's break up this triangular region into r1 and r2.
02:36
For r1, for r1, you have x belongs to 0 to 1.
02:42
And your fx of x in r1 will be y going from 0 to x because the entry for y is 0 and the exit is y equal x.
02:50
3 into 2 minus x into y, d, y, so when you do the integration, you'll be getting 3 into 2 minus x into x square by 2 and this is valid only when x belongs to 0 to 1.
03:02
Now for r2, 1 less than x less than 2, then your fx of x is y going from again 0 to but now it is 2 minus x because the entry for y is 0 but the exit is y is 2 minus x into 3 into 2 minus x into y, d .y.
03:21
So now you will be getting 3 into 2 minus x whole cube divided by 2.
03:27
So that means finally the marginal pdf of x is it looks like this.
03:35
Zero else when x belongs to 0 to 1, 3 into 2 minus x into x squared by 2 this is 0 less than x less than 1.
03:46
3 into 2 minus x whole cube by 2 this is 1.
03:50
3 less than x less than two so this is the marginal pdf of the random variable x now let us find the marginal pdf of y the marginal pdf of y is basically the support of x the joint pdf dx the support of x joint pdf d x obviously x goes from y to 2 minus y 3 into 2 minus x into y d x uh so let's do the integration with respect to x now so 3y i can pull it out what is integration of 2 minus x it is 2 minus x whole square divided by negative because coefficient of x is negative substitute the upper end lower limits so let's see what happens so minus 3y by 2 when you substitute upper 2 minus of 2 minus y is y square minus 2 minus y whole square so i think we can simplify this right so minus t y by two into a square minus b square a minus b into a plus b so y -y -cancel two two cancel so it is minus t y into two y minus two so that means your marginal pdf of y is basically uh six y minus six y square but zero less than y less than one because the support of y is from 0 to 1 and 0 otherwise.
05:24
This is a marginal pdf of y.
05:29
Now the next question, what is the probability that x plus y is less than or equal to 1? this is the question asked.
05:40
So let's plot this region x plus y less than and equal to 1.
05:44
If x plus y is less than equal 1 is a subset of the original region.
05:47
So let me draw the original region again.
05:50
So the original region is this right, this one.
05:54
This is 2...