00:01
In this question, there is given a probability joint answer to function, and this is the function.
00:06
So we have two different variables here.
00:08
One of the variable is x, and the next one is y.
00:11
So in part a, we have to just calculate the value of c here.
00:16
So in order to calculate the value of c, we know that when you just put inside the integral of this value, so we have the double integral here.
00:25
So the double integral should be equal to, let's say, dx, d, y, that should be equal to 1.
00:30
So let's start with the first integral, which is in terms of x here.
00:35
So let's say this is c times.
00:37
This is 3x minus y.
00:39
And the first integral in terms of x, which is 1 .3, and this is the x.
00:45
And then the next integral is that in terms of 4 y.
00:50
And the boundary is between 1 and 2.
00:54
Great.
00:55
Let's first of all, let's calculate the first integral, because inside the integral, we have first the x integral here.
01:02
That should be equal to this is from 1 to 2.
01:05
Let's take the integral of this expression.
01:07
This is c times the integral of 3x in terms of x, which is 3x squared over 2 and minus yx.
01:15
So the boundary is from 1 to 3 and there's a d .y here.
01:20
Let's first of all figure out on the part of this part here.
01:26
And so i can just write as this is from 1 to 2.
01:31
Let's plug in 3 here.
01:33
This is c times.
01:34
Plug in 3.
01:35
That is 27 over 2 minus 3y and minus parentheses.
01:41
This is let's expand the parentheses.
01:43
This is 3 over 2 and plus y.
01:46
I just expand the parentheses with negative sign and this is d .y.
01:50
Let's simplify again inside the parenthesis here, which is c times.
01:56
These are canceled to each other and that is 12 and this one is negative 2y.
02:01
And the y.
02:03
Let's take the integral of this expression with respect to y.
02:07
That should be from one to two.
02:08
Let me just take the integral first.
02:10
This is c times.
02:12
This is 12 y and minus y squared from one to two.
02:17
And the integral of this value should be equal to one because this is the property of, you know, the probability of the probability.
02:24
That's the function.
02:26
And this is c times.
02:27
Let's plug in two first, which is 24, minus four, and minus parentheses.
02:33
Just plug in one.
02:34
This is 12 minus one, which is one here...