please explain 4.1) (10 pts) Suppose that X and Y have the joint probability distribution shown...
4. (20 points) Suppose the joint distribution of X and Y is: fxy(x, y) 1 0 1 2 3 0.04 0.06 0.01 0.00 0.13 0.13 0.02 0.12 0.04 0.06 0.00 0.11 0.07 0.10 0.06 (a) (4 points) Find the marginal distributions of X and Y. (b) (4 points) Given X = 3, what is the probability that random variable Y is at most 2?. (c) (4 points) Are random variables X and Y independent? Why or why not? (d) (4...
Suppose that the following table is the joint probability distribution of two random variables X and Y: х -2 0 2 3 0.27 0.08 0.16 0.2 0.1 0.04 0.1 0.05 a. Find the marginal PDF of X when x=-2, 0, 2, and 3. b. Find the marginal PDF of Y when y=2 and 5. . Find the conditional PDF of x=-2 and 3 given that y=2 has occurred. . Find the conditional PDF of y=2 and 5 given that x=3...
Question 4: Let X and Y be two discrete random variables with the following joint probability distribution (mass) function Pxy(x, y): a) Complete the following probability table: Y 2 f(x)=P(X=x) 1 3 4 0 0 0.08 0.06 0.05 0.02 0.07 0.08 0.06 0.12 0.05 0.03 0.06 0.05 0.04 0.03 0.01 0.02 0.03 0.04 2 3 foy)=P(Y=y) 0.03 b) What is P(X s 2 and YS 3)? c) Find the marginal probability distribution (mass) function of X; [f(x)]. d) Find the...
Suppose the joint probability distribution of two binary random variables X and Y are given as follows. x/y 1 2 0 3/10 0 1 4/10 3/10 X goes along side as 0 and 1, Y goes along top as 1 and 2. a) Show the marginal distribution of X. b) Find entropy H(Y ). c) Find conditional entropy H(X|Y ) and H(Y |X). d) Find mutual information I(X; Y ). e) Find joint entropy H(X, Y ). f) Suppose X...
Consider joint probability distribution given below y fxy (x, у) х 1.0 1 11/32 1/32 1.5 2 1.5 1/4 2.5 4 1/4 3.0 1/8 Determine the following: In your intermediate calculations, round all fractions to three decimal places. Round your answers to three decimal places (e.g 98.765) (a) Conditional probability distribution of Y qiven that X = 1,5. у Fуus 0) 1 2 3 5 (b) Conditional probability distribution of X given that Y 2. 1.0 1.5 2.5 (c) E(YIX...
Suppose the joint probability distribution of two binary random variables X and Y are given as follows. X/Y 1 0 1 2 1 4 0 + 1 (a) Show the marginal distribution of X. [2pts] (b) Find entropy H(Y). [2pts] (e) Find conditional entropy H(XY). (3pts] (d) Find mutual information I(X;Y). [3pts] 2 (e) Find joint entropy H(X,Y). (3pts) Note: The following three proofs are not related to the example in parts (a - e). You need to prove each...
Suppose hat the joint probability distribution of the continuous random variables X and Y is constant on the rectangle 0 < x < a and 0 < y < b for a, b E R+. Show mathematically that X and Y are independent. Hint: (a) Recall JDx "lly f(r, y) dy dx-1 (b) Recall X, Y are independent if ffy fry Suppose hat the joint probability distribution of the continuous random variables X and Y is constant on the rectangle...
1. (10) Suppose the random variables X and Y have the joint probability density function 4x 2y f(x, y) for 0 x<3 and 0 < y < x +1 75 a) Determine the marginal probability density function of X. (6 pts) b) Determine the conditional probability of Y given X = 1. (4 pts)
1. Suppose you have two random variables, X and Y with joint distribution given by the following tables So, for example, the probability that Y o,x - 0 is 4, and the probability that Y (a) Find the marginal distributions (pmfs) of X and Y, denoted f(x),J(Y). (b) Find the conditional distribution (pmf) of Y give X, denoted f(YX). (c) Find the expected values of X and Y, EX), E(Y). (d) Find the variances of X and Y, Var(X),Var(Y). (e)...
The joint distribution of X and Y is given by x/y 1 3 1 0.06 0.42 0.12 2 0.04 0.28 0.08 1. Are X and Y independent or dependent? 2. Prove your answer in part 1.