00:01
So in this question, we're given a joint pdf.
00:05
Fxy is cxy for x in the interval, 0 to 4, and y.
00:17
Sorry, these are open intervals, just to be clear.
00:21
So x in the interval, 0 to 4, and y in the interval 1 to 5, and 0 otherwise.
00:33
So first of all, let's find the value of c.
00:36
So the double integral of cxy, dx, dx, d, y, from x is 0 to 4, and y is 1 to 5, is going to be 1, because that's the total probability.
00:51
So we get c.
00:52
Now the x integral is going to give us a half x squared, so that's a half times 4 squared, which is 8.
00:59
The y integral is going to give us a half y squared.
01:02
So we get a half of 25 minus 1.
01:06
This is equal to 1.
01:09
So c is going to be 2 over 8 times 24, which is 1 over 4 times 24, which is 1 over 96.
01:25
Okay, so now let's find the probability that x is between 1 and 2, and that y is between 2 and 3.
01:40
This is the integral 1 over 96, and we've got x, y, d x, d, y, from x, and y, from x, is 1 to 2 and y is 2 to 3.
01:52
So that's 1 over 96 times a quarter and then we've got 2 squared minus 1, 3 squared minus 2 squared which is 1 over 4 times 96 times 3 times 9 minus 4 which is 5 which gives us 5 over 128.
02:25
So now let's find the probability that x is greater than or equal to 3 and y is less than or equal to 2.
02:32
So we are going to be, we've got 1 over 96, double integral xy d x, from x, 0 to 3 and from y, sorry from x is 3 to 4 i should say, and from y is 1 to 2.
02:52
So this gives us 1 over 96, 1 over 4, 4 squared minus 3 squared, 2 squared minus 1 squared.
03:06
So 1 over 4 times 96 multiplied by 16 minus 9, multiplied by 4 minus 1, which is 3.
03:15
This gives us 7 over 128.
03:20
So let's find the marginal pdfs.
03:23
Fx of x is going to be 1 over 96 x times the integral over 1 .6 x times the integral over 1.
03:30
Y -d -y from y equals 1 to 5, which is 1 over 96 times 2 times 24, which gives us x over 8 for x in the interval 0 to 4, and 0 otherwise.
04:03
Okay, so now the marginal density of y, fy of y, 1 over 96, we can take the y out and then we integrate x from 0 to 4.
04:14
So that's going to be 8 over 96, which is 112.
04:20
So we get y over 12 when y is in the interval 1 to 5 and 0 otherwise.
04:32
So now we want the probability that x plus y is less than 3.
04:38
This is going to be the...
04:45
So for x plus y to be less than 3, we can go from x is 0 to...
04:53
Because y can't be less than one, that means x can't be greater than two.
04:58
So we integrate from x is zero to two, and we integrate from y is zero to three minus x.
05:10
Sorry, y is one to three minus x, d y, and then we've got one over 96 x y in the integral.
05:24
So that's the integral from zero to two d x.
05:27
Now let's do the y integral.
05:28
We're going to get 1 over 96 times 2.
05:33
So 96 times 2 is 192.
05:36
X over 192 and then 3 minus x squared minus 1.
05:45
So it's 1 over 192.
05:47
Integral from 0 to 2 dx.
05:50
And we've got x times 9 minus 6x plus x squared.
05:56
But then we call this minus 1.
05:58
So it's x times 8.
05:59
So we've got 8x, and then we've got minus 6x squared plus x squared.
06:13
So we get 1 over 192, 4x squared minus 2x cubed plus a quarter x to 4 between 0 and 2.
06:25
This is 1 over 192 times 2 squared, which is 4.
06:31
2 cube is 8.
06:32
And then we've got one quarter of 16.
06:41
So these are going to cancel.
06:42
This is four, so we get four over 192, which is 1 over 48.
06:51
So that's the probability that x plus y less than three.
06:58
So now let's find the density function...