(B) If the random variables X and Y have joint density...

(B) If the random variables X and Y have joint density function f(x,y) ={xy/96 0 < x < 4, 1 < y < 5 0 otherwise Find f(x) and f(y) and prove that X and Y are independent , Also find the density function of U = X - 2Y

Transcript

00:01 All right, so we're giving a joint density function for x and y.

00:06 We're going to find the marginal probabilities of x and f of y, and we're going to show that xm are independent.

00:15 So the way we figure out these marginal probabilities is we integrate our function with respect to the opposite variable.

00:22 So here we went f of x.

00:24 So we integrate the function with respect to y on the range of y.

00:28 So what that means, f of x equals the integral of our function, xy over 96, with respect to y, and then we do one to five.

00:47 So that, because we're trying to figure out all these possible, you know, all the one to five range here.

00:52 So we integrate with respect to y.

00:54 So let's go and do that.

00:54 So we get x, then we get y to the second, and then this becomes 192, which we'll show why, because we plus one divide, let's say, yeah, times 96, 2 times 96, and 2 times 96 is 192.

01:13 And then we evaluate this for y equals 1, up to y equals 5.

01:23 And let's see, this is going to give us 25, because 5 squared is 25, 25 x over 192 minus, x over 192 so that's 24 x over and this can be simplified to x over 8 so we're right over here x of x equals x over 8 now let's go ahead and do y so y oh and actually i'm going to include the little subscript for the variable we're talking about x here in the dominoes so it's going to integrate from 0 to 4, the function xy over maybe 6, the x.

02:16 So this becomes x squared y over 192.

02:22 Integrated or evaluated between x is 0 and x is 4.

02:28 The zero is going to kill this.

02:31 So we're just going to have 16 y over maybe 2, which equals y over 12.

02:38 So we get f y of y equals my over 12.

02:44 And you guys see this is capital, capital x, capital y.

02:50 All right.

02:51 So we've done that.

02:52 Oh, and we want to prove the x and y are independent.

02:54 So the independent, if, if f x is x times f big y and y equals f equals a joint.

03:14 Density function.

03:17 So let's go do that.