Let X and Y be two independent random variables having...

Let X and Y be two independent random variables having identical gamma distributions. (a) Find the joint probability density of the random variables U = X / (X + Y) and V = X + Y. (b) Find and identify the marginal density of U.

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Transcript

00:01 Hello everyone, welcome to this video.

00:02 So, here we are given with x, y be an independent random variable and we are given u and v in terms of x and y, we have to find the joint probability density.

00:15 Now, let us solve the first question.

00:18 So, here we have v is equal to x plus y.

00:22 So, our u can be taken as x can be taken as x by x by v.

00:41 So, that is our x will be equal to uv.

00:46 Now, taking the v value.

00:50 So, here we get v is equal to.

00:54 So, in terms of x we have uv.

00:57 So, uv plus y.

01:01 So, taking y aside we get uv v minus uv.

01:08 So, that will be equal to v into 1 minus u.

01:13 So, now, we found in terms of x and in terms of y.

01:18 Now, let us find the joint probability distribution.

01:23 So, for that we have to find the in terms of jacobian.

01:28 So, thus we have jacobian method to solve this.

01:37 So, we write it in terms of dou u by dou x which is equal to.

01:44 So, taking the partial derivative of u with respect to x we get x plus.

01:51 So, u we have x by x plus y the whole square...

Question

Let X and Y be two independent random variables having identical gamma distributions. (a) Find the joint probability density of the random variables U = X / (X + Y) and V = X + Y. (b) Find and identify the marginal density of U.

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