The mean and variance of Y here is computed as:
therefore 37 is the mean of Y here.
The variance of Y here is computed as:
Note that as all Xi are independent, therefore we have
here:
Therefore 51 is the required variance here.
Problem 5. (2 points) If X,, X, ..X40 are independent random variables with means u, =...
9. Let X and Y be independent and identically distributed random variables with mean u and variance o. Find the following: (a) E[(x + 2)] (b) Var(3x + 4) (c) E[(X-Y)] (d) Cov{(X + Y), (X - Y)}
Problem 3: 10 points σ2. Define Assume that U, V, and W are independent random variables with the same common variance X= + W and Y-V-W. 1. Find the variances Var[X] and Var[Y 2. Find the covariance between X and Y, that is: cov [x,Y 3. Find the covariance between (X+Y) and (X - Y), that is: COV[(X +Y), (X -Y)]
number2 how to solve it? Are x1 and x2 independent - yes, they are independent. Random variables X and Y having the joint density 1. 8 2)u(y 1)xy2 exp(4 2xy) fxy (x, y) ux- _ 3 1 1 Undergo a transformation T: 1 to generate new random variables Y -1. and Y2. Find the joint density of Y and Y2 X3)1/2 when X1 and X2 (XR 2. Determine the density of Y are joint Gaussian random variables with zero means...
If the random variables X, Y, and Z have the means ux = 3, uy = -2, and uz = 2, the variances o = 3, o = 3, o2 = 2, the covariances cov(X,Y) = -2, cov(X, Z) = -1, and cov(Y,Z) = 1, U = Y - Z, and V = X - Y +2Z. (a) Find the mean and the variance of U and V, respectively. (b) Find the covariance of U and V.
6.48 Two Gaussian random variables, X and Y, are in- dependent. Their respective means are 4 and 2, and their respective variances are 3 and 5 (a) Write down expressions for their marginal pdfs. (b) Write down an expression for their joint pdf. (c) What is the mean of Z 3X +Y? Z, 3X- Y? (d) What is the variance of Z = 3X + Y? Z, 3X-Y? (e) Write down an expression for the pdf of Z1 3X+Y
Problem D: Suppose X1, .,X, are independent random variables. Let Y be their sum, that is Y 1Xi Find/prove the mgf of Y and find E(Y), Var(Y), and P (8 Y 9) if a) X1,.,X4 are Poisson random variables with means 5, 1,4, and 2, respectively. b) [separately from part a)] X,., X4 are Geometric random variables with p 3/4. i=1
If the random variables X, Y, and Z have the means ji x = 3, My = -2, and uz = 2, the variances of = 3, o = 3, o2 = 2, the covariances cov(X,Y) = -2, cov(X, Z) = -1, and cov(Y,Z) = 1, U = Y - Z, and V = X - Y + 2Z. (a) Find the mean and the variance of U and V. (b) Find the covariance of U and V.
Let X, y, and U be jointly normal zero-mean random variables with variances Problem 1 4, 2, and 1, respectively, such that E XY 1. Assume that U is independent of X and Y Let Z = X + Y + U. Find the joint PDF of X, Y. and Z. Your answer should be explicit C1 and not contain vectors or matrices. Let X, y, and U be jointly normal zero-mean random variables with variances Problem 1 4, 2,...
Use this result without proof: if X and Y are two normal random variables with means ux and My respectively, and variances oź and oſ respectively, and Z = X+Y, Z is also a normal random variable with mean (ux + Hy) and variance (ox +og). a) Suppose Yı, Y2, Yz, Y4 and Y5 are all independent normal random variables, each with a mean of 1 and a variance of 5. What is the probability that (Y1 + 2Y2 +...
Consider two random variables, X and Y. Let E(X) and E(Y) denote the population means of X and Y respectively. Further, let Var(X) and Var(Y) denote the population variances of X and Y. Consider another random variable that is a linear combination of X and Y Z- 3X- Y What is the population variance of Z? Assume that X and Y are independent, which is to say that their covariance is zero.