adding (*) and (**), we have
(b) (2pts) Consider any two random variables X, Y of any distirbution and not necesarily independent....
Consider any two random variables X, Y of any distirbution and not necesarily independent. Given that P(X=a)=p1, P(max(X, Y) =a) =p2, and P(min(X, Y) =a) =p3. Find P(Y=a) in terms of p1, p2 and p3. Hint: Use the first Bayes theorem. Choose a suitable event-complement pair to partition the probability space by comparing X and Y .
(a) (2 pts) An urn contains 3 red and 5 green balls. At each step of this game, we pick one ball at random, note its color and return the ball to the urn together with anoter ball of the same color. Prove by induction that the probability that the ball we pick a red ball at the n-th step is 3/8. (b) (2pts) Consider any two random variables X, Y of any distirbution and not necesarily independent. Given that...
Problem 2. (6 pts) Independence and Conditional Probability (a) (2 pts) An urn contains 3 red and 5 green balls. At each step of this game, we pick one ball at random, note its color and return the ball to the urn together with anoter ball of the same color. Prove by induction that the probability that the ball we pick a red ball at the n-th step is 3/8. (b) (2pts) Consider any two random variables X, Y of...
33. Let X and Y be independent exponential random variables with respective rates λ and μ. (a) Argue that, conditional on X> Y, the random variables min(X, Y) and X -Y are independent. (b) Use part (a) to conclude that for any positive constant c E[min(X, Y)IX > Y + c] = E[min(X, Y)|X > Y] = E[min(X, Y)] = λ+p (c) Give a verbal explanation of why min(X, Y) and X - Y are (unconditionally) independent. 33. Let X...
1. Let X and Y be two independent exponential random variables with parameters λ and μ, respectively. Compute the probability P(X Y| min(X,Y)-x).
please help me! thanks 3, (20%) Prove that if any two of the three random variables X, Y, and Z are independent, I(X; Y) I(X; Y1Z) holds. 3, (20%) Prove that if any two of the three random variables X, Y, and Z are independent, I(X; Y) I(X; Y1Z) holds.
2. Let X and Y be two independent discrete random variables with the probability mass functions PX- = i) = (e-1)e-i and P(Y = j-11' for i,j = 1, 2, Let {Uni2 1} of i.i.d. uniform random variables on [0, 1]. Assume the sequence {U i independent of X and Y. Define M-max(UhUn Ud. Find the distribution
Problem 6: 10 points Assume that X and Y are independent random variables uniformly distributed over the unit interval (0,1) 1. Define Z max (X. Y) as the larger of the two, Derive the C.DF. and density function for Z. 2. Define W min(X,Y) as the smaller of the two. Derive the C.D.F.and density function for W 3. Derive the joint density of the pair (W. Z). Specify where the density if positive and where it takes a zero value....
Comparing two densities. Joint density (a) for random variables X and Y is given by: fxy(x, y) = 6e-23-if 0 <y<I<0. Joint density (b) for random variables X and Y is given by: fxY(I, y) = 2e -2- if 0 <1,7 <00. Fill in the following chart and determine whether or not X and Y are independent for both densities (a) and (b). fx() fy(y) EX EY EXY Cou(X,Y) Independent?
4. Let X and Y be independent standard normal random variables. The pair (X,Y) can be described in polar coordinates in terms of random variables R 2 0 and 0 e [0,27], so that X = R cos θ, Y = R sin θ. (a) (10 points) Show that θ is uniformly distributed in [0,2 and that R and 0 are independent. (b) (IO points) Show that R2 has an exponential distribution with parameter 1/2. , that R has the...