Question

Let X be a discrete r.v. and Y be a continuous r.v. such that the conditional...

Let X be a discrete r.v. and Y be a continuous r.v. such that the conditional distribution of X given Y = y is a (discrete) geometric distribution with probability for success p, and such that Y has pdf f_Y(y) = 3y for 0 < y < 1 (and zero otherwise).

a) Compute the pmf of X.

b) Compute E[X].

c) Does the r.v. Var(X | Y) have a finite expectation?

Math   Statistics-And-Probability   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

The PDF given in question f_Y(y) = 3y for 0 < y < 1 is wrong.

I take it as fY(y) = 3y2; 0 < y < 1

Given X is a discrete RV and Y be a continuous RV such that the conditional distribution of X given Y =y is geometric distribution with probability for success P.

So, the conditional PMF is

P (X = x | Y = y) = (1 – p)x p; x = 0,1,2,3…

So, X,Y are independent RVs.

Also, the PDF of Y is fY(y) n= 3y2; 0 < y < 1

The conditional expectation is E (X | Y) = (1 - p) / p

The joint mixture PDF is,

P (X = x, y) = P (X = x | Y = y) fY(y)

P (X = x, y) = (1 – p)xp(3y2)

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Let continuous random variables X, Y be jointly continuous, with the following joint distribution fXY​(x,y) =...
Let continuous random variables X, Y be jointly continuous, with the following joint distribution fXY​(x,y) = e-x-y ​for x≥0, y≥0 and fXY(x,y) = 0 otherwise​. 1) Sketch the area where fXY​(x,y) is non-zero on x-y plane. 2) Compute the conditional PDF of Y given X=x for each nonnegative x. 3) Use the results above to compute E(Y∣X=x) for each nonnegative x. 4) Use total expectation formula E(E(Y∣X))=E(Y) to find expected value of Y.
Let X, Y be two random variables with a joint pmf f(x,y)=(x+y)/12 x=1,2 and y=1,2 zero...
Let X, Y be two random variables with a joint pmf f(x,y)=(x+y)/12 x=1,2 and y=1,2 zero elsewhere a)Are X and Y discrete or continuous random variables? b)Construct and joint probability distribution table by writing these probabilities in a rectangular array, recording each marginal pmf in the "margins" c)Determine if X and Y are Independent variables d)Find P(X>Y) e)Compute E(X), E(Y), E(X^2) and E(XY) f)Compute var(X) g) Compute cov(X,Y)
Let X and Y be discrete random variables, their joint pmf is given as Px,y =...
Let X and Y be discrete random variables, their joint pmf is given as Px,y = ?(? + ? + 2)/(B + 2) for 0 ≤ X < 3, 0 ≤ Y < 3 Where B=2. a) Find the value of ? b) Find the marginal pmf of ? and ? c) Find conditional pmf of ? given ? = 2
Let f (x, y) = c, 0 ≤ y ≤ 4, y ≤ x ≤ y...
Let f (x, y) = c, 0 ≤ y ≤ 4, y ≤ x ≤ y + 1, be the joint pdf of X and Y. 1. Determine h(y | x), the conditional pdf of Y, given that X = x. 2. Determine g(x | y), the conditional pdf of X, given that Y = y. 3. Compute E(Y | x), the conditional mean of Y, given that X = x. 4. Compute E(X | y), the conditional mean of...
SOLUTION REQUIRED WITH COMPLETE STEPS Let X and Y be discrete random variables, their joint pmf...
SOLUTION REQUIRED WITH COMPLETE STEPS Let X and Y be discrete random variables, their joint pmf is given as Px,y = ?(? + ?)/(B + 1) for 0 < X ≤ 3, 0 < Y ≤ 3 (Where B=5) a) Find the value of ? b) Find the marginal pmf of ? and ? c) Find conditional pmf of ? given ? = 2
SOLUTION REQUIRED WITH COMPLETE STEPS Let X and Y be discrete random variables, their joint pmf...
SOLUTION REQUIRED WITH COMPLETE STEPS Let X and Y be discrete random variables, their joint pmf is given as Px,y = ?(? + ? + 1)/(B + 1) for 0 ≤ X < 3, 0 ≤ Y < 3 (Where B=7) a) Find the value of ? b) Find the marginal pmf of ? and ? c) Find conditional pmf of ? given ? = 1
Let X and Y be discrete random variables, their joint pmf is given as ?(x,y)= ?(?...
Let X and Y be discrete random variables, their joint pmf is given as ?(x,y)= ?(? + ? − 2)/(B + 1) for 1 < X ≤ 4, 1 < Y ≤ 4 Where B is the last digit of your registration number ( B=3) a) Find the value of ? b) Find the marginal pmf of ? and ? c) Find conditional pmf of ? given ? = 3
Let X and Y be two continuous random variables with joint probability density function f(x,y) =...
Let X and Y be two continuous random variables with joint probability density function f(x,y) = xe^−x(y+1), 0 , 0< x < ∞,0 < y < ∞ otherwise (a) Are X and Y independent or not? Why? (b) Find the conditional density function of Y given X = 1.(
Complete the following: (a) Var (X) ≥ 0 for any discrete r.v. X. Also, explain why...
Complete the following: (a) Var (X) ≥ 0 for any discrete r.v. X. Also, explain why X is constant if and only if Var (X) = 0. Hint: write Var (X) = E (X − µ)2 = Px (x − µ)2 pX (x), where µ = E (X). (b) Let X represent the dierence between the number of heads and the number of tails obtained when a fair coin is tossed n times. Find E (X) and Var (X)
Let X and Y be two continuous random variables with joint probability density function f(x,y) =...
Let X and Y be two continuous random variables with joint probability density function f(x,y) = 6x 0<y<1, 0<x<y, 0 otherwise. a) Find the marginal density of Y . b) Are X and Y independent? c) Find the conditional density of X given Y = 1 /2
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT