2. Let X be a gamma random variable with parameters α = 2 and β = (1)/(8) and given X the random

Step 1: Since Y is uniformly distributed between 2X and 8X, the probability density function of

2. Let X be a gamma random variable with parameters α = 2...

2. Let X be a gamma random variable with parameters $\alpha = 2$ and $\beta = \frac{1}{8}$ and given X the random variable Y is uniformly distributed between 2X and 8X. Find $P(X > \frac{3}{8}|Y = 1)$.

Transcript

00:01 So in this question we're told that x is a gamma variable with parameters alf equals 2 and beta equals 1 8th.

00:12 And y given x is uniform between 2x and 8x.

00:21 Then we need to find the expected value of y.

00:25 So expected value of y is the expected value of the expected value of y given x by the law of total probability.

00:32 So that's the expected value of the expected value of y given x, which is just going to be the integral of 1 over 6x times y, d .y from 2x to 8x.

00:51 So that's the expected value of a half y squared.

00:56 So that's 1 over 12 times, well, 1 over 12x times 8x times 8x.

01:05 Squared minus 2x squared.

01:10 So we get 8 squared minus 2 squared divided by 12 times x so that gives us the expected value of 5x.

01:20 So that's 5 times the expected value of x and the expected value of the gamma variable is just alpha over beta.

01:30 So that's 5 alpha over beta and we have alpha is 2, beta is 8.

01:36 So we have 2 times 8.

01:38 Is alpha over beta times 5 gives us 80.

01:43 So that's the expected value of y.

01:48 And for part b, we want to find f of x given y, and that's going to be f of x and y divided by f of y.

02:02 Or we can say that this is, so we can say that this is f of y given x, f of x, f of x over f of y.

02:13 But first we need to find f of y.

02:20 So f of y given x is equal to 1 over 6x for y between 2x and 8x.

02:32 So f of y is going to be, or f of y and x, is equal to f of y given x times f of x.

02:47 So that's going to be 1 over 6x, f of x, and f of x is beta to the alpha over gamma alpha.

02:57 So we have 1 over 6x, beta to the alpha over gamma alpha, x to the alpha minus 1, e to the minus beta x.

03:11 And we have beta is 1 eighth and alpha equals 2...

Question

2. Let X be a gamma random variable with parameters $\alpha = 2$ and $\beta = \frac{1}{8}$ and given X the random variable Y is uniformly distributed between 2X and 8X. Find $P(X > \frac{3}{8}|Y = 1)$.

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