5. [10 points] The random variables X and Y have a joint...

5. [10 points] The random variables X and Y have a joint density function given by $f(x,y) = 3e^{-3x}/x$, for $0 < x < \infty$, $0 < y < x$. (a) Find $f_{Y|X}(y|x)$. (b) Find $Var(Y|X = 1)$. (c) Find $Var(Y)$.

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Transcript

00:01 In this problem, we are given the following joint density probability function.

00:07 F of x, y equal to c times x times 1 plus y when x is between 0 and 2 and y is between 0 and 6 and it is 0 otherwise.

00:24 Okay, we have three parts so let's get started with the first one.

00:29 In the first part, we will compute, we will find the value of this constant c and we will obtain it using the normalization condition.

00:39 So if we integrate our function over the entire space, we should get 1.

00:47 So restricting ourselves to the region where our function is non -zero, we have integral from 0 to 2 dx, integral from 0 to 6 dy, c times x times 1 plus y equal to 1.

01:04 Okay, these will be all elementary integrals so let me just write down the result.

01:08 For the y integral, we have 24 x times c and for the x integral, we have 48 c.

01:19 Therefore, we see that c should be equal to 1 over 48.

01:28 In the next part, we will compute the probability that x is less than or equal to 1 and y is less than or equal to 1.

01:40 Okay, we have integral from minus infinity to 1 dx, integral from minus infinity to 1 dy, f of x y.

01:51 And again, considering only the region where our function is non -zero, we have integral from 0 to 1 dx, integral from 0 to 1 dy, c times x times 1 plus y.

02:07 Okay, again i will just write down the results for these integrals...

Question

5. [10 points] The random variables X and Y have a joint density function given by $f(x,y) = 3e^{-3x}/x$, for $0 < x < \infty$, $0 < y < x$. (a) Find $f_{Y|X}(y|x)$. (b) Find $Var(Y|X = 1)$. (c) Find $Var(Y)$.

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