Suppose that X and Y have the following joint probability density function. f(x,y) = 134² 0 < x < 5, y > 0, x − 2 < y < x+2 (a) Find E(XY). (b) Find the covariance between X and Y.
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- Fill out the table giving the joint and marginal PMFs for X and Y. Find E[X] and E[Y]. Find the covariance of X and Y. Are X and Y independent?Suppose that X and Y have the following joint probability density function. 3 X, 0 0, x −4 < y < x +4 400 f(x, y) = (a) Find E(XY). (b) Find the covariance between X and Y.Suppose that X and Y have the following joint probability density function. f(x, y) = ₁34y, 0 0, x − 2 < y < x + 2 154, (a) Find E(XY). (b) Find the covariance between X and Y. Round your answer to 4 decimals. Round your answer to 4 decimals.
- Suppose that X and Y have the following joint probability density function.f (x, y) = 3x 400 0 < x < 6, y > 0, x − 4 < y < x + 4 (a) Find E(XY). (b) Find the covariance between X and Y.Value of Y 14 22 30 40 65 0.02 0.05 0.10 0.03 0.01 Value of X 5 0.17 0.15 0.05 0.02 0.01 8 0.02 0.03 0.15 0.10 0.09The random variable Xi , i = 1, 2, models the proportion of type i switches on a panel that are turned off during a training exercise. The joint probability density function of X1 and X2 is (attached): Find the Covariance of (X1 , X2).
- C2. Let X and Y be random variables, and let a and b be constants. (a) Starting from the definition of covariance, show that Cov(aX, Y): = a Cov(X, Y). You may find it helpful to remember that if EX = µx, then EaX = αμχ· (b) Show that Cov(X + b, Y) = Cov(X, Y). Now let X, Y, Z be independent random variables with common variance o². (c) Find the value of Corr(2X - 3Y + 4, 2Y – Z - 1). You may use any facts about covariance from the notes, including those from parts (a) and (b) of this question, provided you state them clearly.= B1. Let X₁ and Y; be random variables with Var(X;) = o² and Var(Y₂) o for all ie {1,...,n}. Assume that each pair (X₁, Y;) has correlation Corr(X, Y) = p, but that (X₁, Y₂) and (X₁, Y;) are independent for all i j. (a) What is Cov(X₁, Yi) in terms of Ox, Oy (b) Show that Cov(X₁,Y) = poxoy/n, where Y is the average of the Y₁. (c) Determine Cov(X, Y). and p?Let 3x2, 0Suppose X and Y have joint probability density function f(x, y) = { 0(a C(x² + y²), if 0E(XY) – E(X)E(Y) 1. The correlation of X and Y is defined as px,y if X and Y are independent, then px,y = 0. Quickly show that o(X)o(Y) 2. Let X be a random variable with P(X = k) = | for k = -1,0, 1. Let Y = X². (a) What is the p.m.f. of Y? (b) Are X and Y independent? (c) Calculate E(XY) – E(X)E(Y). (d) True or false (for any random variables X and Y): X and Y are independent if and only if E(XY) – E(X)E(Y). 3. An exam consists of twenty multiple choice questions. Each question has five different possible answers, only one of which is correct. A person randomly guesses on each question. What is the variance of the number of questions that are correct? 4. Suppose ten people get on an elevator. There are eight buttons representing floors on this elevator. Each person presses a random button. The elevator will only stop at a floor if at least one person pressed that floor's button. What is probability the elevator stops at exactly six floors? What is the variance of the number of floors…Let X be a continuous random variable symmetric about Y. Let Z = 1 if X >Y OR Z = 0 if X <= Y. Find the covariance of |X| and Z.SEE MORE QUESTIONSRecommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON
A First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON