00:01
Okay, so we have the joint probability density function of two continuous random variables is given by this function.
00:09
So we have that it's equal to c some constant times 3x minus y.
00:13
If x is between 1 and 3 and y is between 1 and 2 and 0 otherwise, and part a of this question, we want to find what c is.
00:22
So the point here is because this is a density function, we need to integrate it over the domain, and we need the integral to be equal to 1.
00:30
So we need the integral over r2 of f of x y d x, d y, to be equal to 1.
00:40
For this to be a valid probability density, or in other words, the integral from minus infinity to infinity, minus infinity to infinity, f of x y, d x, d x, d y, equals 1.
00:55
So we need this.
00:57
So all we need to do now is plug in our function.
00:59
So although these are infinite, integrals this function is only uh only non -zero on these intervals here so the x integral is actually going to be so the x integral is going to be from one to three so the inner integral here one to three the eight integral the y integral is going to be between one and two so this is just one to two of c times three x minus y d x d y d x and we need this to be be equal to 1.
01:34
So from here, all we have to do is integrate this.
01:37
So if we do the first integral, the inner integral, what do we get? so we get that if we leave the 80 integral alone and take the c8 front, and we need to integrate this over x, we get 3x squared over 2 minus yx between 1 and 3 dy equals 1.
02:05
So now we need to plug the numbers in.
02:07
So we have c times the integral from 1 to 2, 3 over 2 times 3 squared is going to be 9.
02:16
So 3 squared minus 1 squared, which is 1 minus y times 3 minus 1.
02:30
Right, dy equals 1 now.
02:33
This implies that if we just work these, things eight this integral so this is going to be nine minus one is eight eight divided by two is four four times three is twelve so this is going to be 12 minus two y d y equals one now we need to do this y integral so this gives us that uh 12 y minus two y squared over two between two and one if we just plug everything in so these two is cancel.
03:10
We're going to get c times 12, 2 minus 1 minus 2 squared, 4 minus 1 equals 1.
03:22
If we just work everything 8, 2 minus 1 is 1.
03:25
So this is 12, so this is c times 12 minus 4 minus 1 is 3.
03:31
So 12 minus 3 is 9...