please write step by step and clear extend answer (d) Consider two random variables X and...
2. Let X and X be two random variables with the following joint PMF Yix 2 0 2 0 0.1 0.05 0.05 0.15 0.1 0.05 0.1 0.05 0.05 0.05 4 0.05 0.05 0.02 0.1 0.03 total 0.2 0.2 0.12 0.3 0.18 total 0.45 0.3 0.25 1 1) Find E[X] and E[Y]. (10 points) 2) What is the covariance of X and Y? (20 points) 3) Are X and Y independent? Explain. (10 points)
1. Consider a discrete bivariate random variable (X,Y) with the joint pmf given by the table: Y X 1 2 4 1 0 0.1 0.05 2 0.2 0.05 0 4 0.1 0 0.05 8 0.3 0.15 0 Table 0.1: p(, y) a) Find marginal distributions of X and Y, p(x) and pay respectively. b) Find the covariance and the correlation between X and Y.
The discrete random variables ? and ? have joint probability function ?, where ? is given by the following table: X 1 2 3 4 1 0.1 0.2 0.1 0.05 Y 2 0.05 0 0.1 0.1 3 0 0.2 0.05 0.05 a) Determine ?(1 < ? ≤ 3, 1 ≤ ? ≤ 2). [4 marks] b) Calculate ?(?^2 ?). [4 marks] c) Find the marginal probability functions ? and ℎ of ? and ? respectively. [4 marks] d) Are ?...
Consider two random variables X and Y be the proportion of time that a person travels to work by bus and MTR respectively (there are other modes as well). Moreover, X and Y has the following joint distribution f(x,y) 24xy, 0 x f(x,y)-0 otherwise 1,0 y 1, x + y 1 () Find the marginal distributions g(x) and h(y) respectively. (ii) Find the conditional density function fxly) (ii Ifit is known that a person has 0.75 chance of using bus,...
Please show work step by step 3) Two random variables X and Y have the following joint PDF fx(x, y) = 1zu(x)u(y) e “CI)-(3) Calculate: a) P (2<x<4, -1<Y<5); b) P (0<x< 00, -00<Y<-2).
Consider two discrete random variables X and Y with joint p.m.f given below . 0.200.15 -1 0 0.05 0.15 0.10 1 0.200.15 Find the joint p.m.f. of U-X Y and V X Yand enter it below Pulu,y) -1 0 -1 0 Chec
Problem 8.2 X Y Discrete random variables X, Y have joint pmf given in the table to the right, where X takes values in {1,2,3,4} and Y takes values in {1,2,3). 2 3 1 2 3 0. 100.3 0 0.2 0.1 0 0.05 0.1 0 0.1 0.05 (e) Compute the MAP estimate of X given the observation Y = 2. Compute the posterior probabiity of error of this estimate, given that Y = 2. (f) Compute the MMSE estimate of...
1. Consider two random variables X and Y with joint density function f(x, y)-(12xy(1-y) 0<x<1,0<p<1 otherwise 0 Find the probability density function for UXY2. (Choose a suitable dummy transformation V) 2. Suppose X and Y are two continuous random variables with joint density 0<x<I, 0 < y < 1 otherwise (a) Find the joint density of U X2 and V XY. Be sure to determine and sketch the support of (U.V). (b) Find the marginal density of U. (c) Find...
. Suppose we have the following joint distribution for random variables X and Y 2 0.1 0.2 0.1 4 0 0.3 0.1 6 0 0 0.2 (a) Find p(X). That is find the marginal distribution of X. (b) Find p(Y). That is find the marginal distribution of Y (c) Find the distribution of X conditional on Y = 3. (d) Find the distribution of X conditional on Y 2 (e) Are X and Y independent? You should be able to...
Suppose X, Y are random variables whose joint PDF is given by . 1 0 < y < 1,0 < x < y y otherwise 0, 1. Find the covariance of X and Y. 2. Compute Var(X) and Var(Y). 3. Calculate p(X,Y).