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4. Suppose X and Y are independent random variables with the same probability distribution, given by...
Suppose the joint probability distribution of two binary random variables X and Y are given as...
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 the joint probability distribution of two binary random variables X and Y are given as...
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...
4. Suppose that the joint pdf of the random variables X and Y is given by...
4. Suppose that the joint pdf of the random variables X and Y is given by f(x, y) = cx^2 + xy 3 , if 0 < x < 1, 0 < y < 2 0, otherwise. (a) Find the constant value (b) Find the marginal pdf of X. Include the support. (c) Find the conditional density function Y given X = x, i.e., f(y|x) (d) Find the conditional expectation E(Y |X = x). (e) Are X and Y independent?...
5. Suppose we have two random variables X and Y. They are discrete and have the...
5. Suppose we have two random variables X and Y. They are discrete and have the exact same distribution and also independent. You see below the distribution of X which of course also the distribution of Y as well, that is what we called independent and identically distributed) P(X =- X. Remem- a./ (-) Find and draw the cumulative distribution function F() function of ber that F(x) -P(X S) HINT: For the next 3 parts you might want to make...
Part 2. Random Variables 4. Two independent random variables Xand y are given with their distribution laws 0.3 0.7 0.8...
Part 2. Random Variables 4. Two independent random variables Xand y are given with their distribution laws 0.3 0.7 0.8 0.2 Pi Find the distribution law and variance for the random variable V-3XY 5. There are 7 white balls and 3 red balls in a box. Balls are taken from the box without return at randomm until one white ball is taken. Construct the distribution law for the number of taken balls. 6. Let X be a continuous random variable...
4 Suppose that W, X, Y, Z are independent random variables, each with probability density function...
4 Suppose that W, X, Y, Z are independent random variables, each with probability density function f(t) -4t3,0st s 1. Find. (b) fw.x.y.z(w, x, y, z) (c) Fxy (x,y)
Suppose that X and Y are independent uniform distribution over interval [0,1] random variables. Find the...
Suppose that X and Y are independent uniform distribution over interval [0,1] random variables. Find the probability density function of the product W= XY .
Suppose hat the joint probability distribution of the continuous random variables X and Y is cons...
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...
Suppose X and Y are jointly continuous random variables with probability density function
Suppose X and Y are jointly continuous random variables with probability density function f(х+ у)={1/6(x + y), 0 < х < 1, 0 < у < 3; 0 , else} a) Find E[XY]. b) Are X and Y independent? Justify your answer citing an appropriate theorem.
Let random variables X and Y have the bi-variate exponential CDF (cumulative distribution function) : F(x,y)...
Let random variables X and Y have the bi-variate exponential CDF (cumulative distribution function) : F(x,y) = 1 - exp(-x) - exp(-y) + exp(-x-y-xy) Given x > 0, y>0 a) Determine the probability that 4 < X given that Y = 2 b) Determine the probability that 4 < X given that Y is less than or equal to 2
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...
5. Suppose we have two random variables X and Y. They are discrete and have the exact same distribution and also independent. You see below the distribution of X which of course also the distribution of Y as well, that is what we called independent and identically distributed) P(X =- X. Remem- a./ (-) Find and draw the cumulative distribution function F() function of ber that F(x) -P(X S) HINT: For the next 3 parts you might want to make...
Part 2. Random Variables 4. Two independent random variables Xand y are given with their distribution laws 0.3 0.7 0.8 0.2 Pi Find the distribution law and variance for the random variable V-3XY 5. There are 7 white balls and 3 red balls in a box. Balls are taken from the box without return at randomm until one white ball is taken. Construct the distribution law for the number of taken balls. 6. Let X be a continuous random variable...
4 Suppose that W, X, Y, Z are independent random variables, each with probability density function f(t) -4t3,0st s 1. Find. (b) fw.x.y.z(w, x, y, z) (c) Fxy (x,y)
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...
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