Assume that X and Y are discrete random variables having the joint pmf given by the following chart
Y
0 1 2
0 0.1 0.1 0.3
X 1 0.3 0.1 0.1
a. Find the probability that Y is greater than X.
b. Find the covariance between X and Y.
Assume that X and Y are discrete random variables having the joint pmf given by the...
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.
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...
12) Random variables X & Y have joint pmf given in the table. Y = 1 Y = 2 Y= 3 X = 1 0.3 0.1 0 X = 2 0.1 0.3 0.2 In problem (12), determine Var(X | Y = 3) a) 2.4 b) 2.0 c) 1.4 d) .8 e) 0
12) Random variables X & Y have joint pmf given in the table. Y = 1 Y = 2 Y= 3 X = 1 0.3 0.1 0 X = 2 0.1 0.3 0.2 In problem (12), determine E(3Y + 1.2 | X < Y ) a) 9.0 b) 9.2 c) 9.5 d) 9.8 e) 10.0
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)
13. Let X and Y be rv's whose joint PMF is given by: Y=1 2 3 X=0 0.2 0.1 0 1 0.1 0.3 0. 2 . 0 0 0.3 Compute the covariance and correlation matrix of the random vector (X,Y).
6 X and Y are two discrete random variables with the following PMF. IN IN IA. a. | Find the marginal pmf's for X and Y. b. Draw the joint CD c. Calculate the probability of the events: A-(X>0), B (xeY), and C-X Y for the 3 pt 3 pt. indicated PMF t. Are X, Y independent? Prove. 2 pt. t.
6. (20%) Consider two random variables X and Y with the joint PMF given in Table 2. Table 2: Joint PMF of X and Y Y =0Y 1 X 0 X 1 0 (a) (5%) Find the PMF of X and PMF of Y. (b) (5%) Find EX, EY, Var(X), Var(Y (c) (10%)Find the MMSE estimator of X given Y, (M) for both Y 0 and Y 1
ciule jolh! PMF and the marginal PMFs? 6.14 Let X and Y be discrete random variables. Show that the function p: R2 R defined by p(r, y) px(x)pr(y) is a joint PMF by verifying that it satisfies properties (a)-(c) of Proposition 6.1 on page 262. Hint: A subset of a countable set is countable CHAPTER SIX Joindy Discrete Random Variables 6.2 Joint and marginal PMFs of the discrete random variables x numher of bedrooms and momber of bwthrooms of a...
3. Let f(x,y) = xy-1 be the joint pmf/ pdf of two random variables X (discrete) and Y (continuous), for x = 1, 2, 3, 4 and 0 <y < 2. (a) Determine the marginal pmf of X. (b) Determine the marginal pdf of Y. (c) Compute P(X<2 and Y < 1). (d) Explain why X and Y are dependent without computing Cou(X,Y).