5. Prove the following identity: ???(?, ?) = ?(??) − ????, where cov(X,Y) is the covariance between random variable X and Y, ?? is the mean of X and ?? is the mean of Y.
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5. Prove the following identity: ???(?, ?) = ?(??) − ????, where cov(X,Y) is the covariance...
Let X and Y be two random variables such that: Var[X]=4 Cov[X,Y]=2 Compute the following covariance: Cov[3X,X+3Y]
Prove cov(x+y,x-y) where x and y are independent.
1) Let X and Y be random variables. Show that Cov( X + Y, X-Y) Var(X)--Var(Y) without appealing to the general formulas for the covariance of the linear combinations of sets of random variables; use the basic identity Cov(Z1,22)-E[Z1Z2]- E[Z1 E[Z2, valid for any two random variables, and the properties of the expected value 2) Let X be the normal random variable with zero mean and standard deviation Let ?(t) be the distribution function of the standard normal random variable....
Prove the following properties using the definition of the variance and the covariance: Q1. Operations with expectation and covariances Recall that the variance of randon variable X is defined as Var(X) Ξ E [X-E(X))2], the covariance is Cov(X, ) EX E(X))Y EY) As a hint, we can prove Cov(aX + b, cY)-ac Cov(X, Y) by ac EX -E(X)HY -E(Y)ac Cov(X, Y) In a similar manner, prove the following properties using the definition of the variance and the covariance: (a) Var(X)-Cov(X,...
5. The means, standard deviations, and covariance for random variables X, Y, and Z are given below. Ux= 3, uy = 5, uz = 7 Ox= 1, OY = 3, oz = 4 cov(X, Y) = 1, cov (X, Z) = 3, and cov (Y,Z) = -3 T= X-28 +3 Z var(T) = 16. For a random variable X with an unknown distribution. The mean of X is u = 22 and tting a randomly chosen value of X
4. Recall that the covariance of random variables X, and Y is defined by Cov(X,Y) = E(X - Ex)(Y - EY) (a) (2pt) TRUE or FALSE (circle one). E(XY) 0 implies Cov(X, Y) = 0. (b) (4 pt) a, b, c, d are constants. Mark each correct statement ( ) Cov(aX, cY) = ac Cov(X, Y) ( ) Cor(aX + b, cY + d) = ac Cov(X, Y) + bc Cov(X, Y) + da Cov(X, Y) + bd ( )...
6. (a) State the definition of the covariance Cov(x,Y) of two random variables X and Y. (b) Consider the two continuous random variables X and Y of Ques- tion 2. with joint density f(x, y) otherwise i. Find μχ.y the expectations of X, Y respectively.
Explain how the covariance between a variable X and a constant b, Cov ( X, b ), is equal to zero. (Hint: you may want to use the covariance formula)
5. Suppose the random variables X, Y and Z are related through the model Y = 2 + 2X + Z, where Z has mean 0 and variance o2 = 16 and X has variance of = 9. Assume X and Z are independent, the find the covariance of X and Y and that of Y and Z. Hint: write Cov(X, Y) = Cov(X, 2+2X+Z) and use the propositions of covariance from slides of Chapter
5. Suppose the random variables X, Y and Z are related through the model Y = 2 + 2X + 2, where Z has mean 0 and variance o2 = 16 and X has variance of = 9. Assume X and Z are independent, the find the covariance of X and Y and that of Y and Z. Hint: write Cov(X,Y) = Cov(X, 2+2X+Z) and use the propositions of covariance from slides of Chapter