2. Given the random variables x and y with joint probability distribution X/Y - 4 2...
Given the following joint distribution of two random variables X
and Y
(a) Compute marginal distribution PX(x)
(b) Compute marginal distribution PY(y)
(c) What is the conditional probability P(Y | X = 2)?
20.10 0.05 0.15 0.10 0.10 4 0.04 0.02 0.06 0.04 0.04 6 0.04 0.02 0.06 0.06 0.02 8 0.02 0.01 0.03 0 0.04
The discrete random variables X and Y take integer values with joint probability distribution given by f (x,y) = a(y−x+1) 0 ≤ x ≤ y ≤ 2 or =0 otherwise, where a is a constant. 1 Tabulate the distribution and show that a = 0.1. 2 Find the marginal distributions of X and Y. 3 Calculate Cov(X,Y). 4 State, giving a reason, whether X and Y are independent. 5 Calculate E(Y|X = 1).
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...
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...
If the joint probability distribution of three discrete random variables X, Y , and Z is given by: f(x, y, z) = (x + y)z / 63 , for x = 1, 2; y = 1, 2, 3; z = 1, 2. Find the probability P(X = 2, Y + Z ≤ 3)
Consider the following joint probability distribution on the random variables X and Y given in matrix form by Pxy P11 P12 P13 PXY-IP21 p22 p23 P31 P32 P33 P41 P42 P43 HereP(i, j) P(X = z n Y-J)-Pu represents the probability that X-1 and Y = j So for example, in the previous problem, X and Y represented the random variables for the color ([Black, Red]) and utensil type (Pencil,Pe pblackpen P(X = Black Y = Pen) = P(Black n...
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...
ka (3) [6 pts] X and Y are discrete random variables with the following joint distribution: 14 22 30 065 102 0.05 0.10 0.03 0.01 Value ofX 50.17 0.15 0.05 0.02 0.01 8 0.02 0.03 0.15 0.10 (a) Calculate the probability distribution, mean, and variance of Y (b) Calculate the prohability distribution, mea, and variane of Y given X (c) Calculate the covariance and correlation between X and Y 8
The joint probability density function for continuous random variables X and Y is given below. f (x) = x + y, 0 < x < 1, 0 < y < 1 if; 0, degilse. (a) Show that this is a joint density function. (b) Find the marginal density of X . (c) Find the marginal density of Y . (d) Given Y = y find the conditional density of X . (e) P ( 1/2 < X < 1|Y =...