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Page 13 of 13 15. (3 points each) Let X be a random variable with a...
3. Let X be a discrete random variable with the following PMF: 0.1 for x 0.2 for 0.2 for x=3 Pg(x)=〈 0.1 for x=4 0.25 for x=5 0.15 for x=6 otherwise a) (10 points) Find E[X] b) (10 points) Find Var(X) c) Let Y-* I. (15 points) Find E[Y] II. (15 points) Find Var(Y) X-HX 4. Consider a discrete random variable X with E [X]-4x and Var(X) = σ. Let Y a. (10 points) Find E[Y] b. (20 points) Find...
3. Let X be the height of Zebras, assume the X is a random variable with mean 10 and variance 20. Suppose Y is be the weight of Zebras, assume the Y is a random variable with mean 10 and variance 40. Let E(XY)-80 (a) Find the covariance and correlation between X and Y. Find the covariance and correlation between aX + b and cY + d. a,b,c, and d are unknown constants. Your answer can depend on them. (b)...
7. X is a random variable with a mean of 2 and a variance of 3, and Y is a random variable with a mean of 4 and a variance of 5, and the covariance between X and Y is -3. Define (a) Find the expected value of W. b) Find the variance of W
Problem 1. 15 points] Let X be a uniform random variable in the interval [-1,2]. Let Y be an exponential random variable with mean 2. Assunne X and Y are independent. a) Find the joint sample space. b) Find the joint PDF for X and Y. c) Are X and Y uncorrelated? Justify your answer. d) Find the probability P1-1/4 < X < 1/2 1 Y < 21 e) Calculate E[X2Y2]
4. [-14 Points] DETAILS (4pt) The variance of random variable X is 1 and the variance of random variable Y is 4. The correlation coefficient between the two random variables X and Y is 0.2. (a) (1pt) Find the covariance between X and Y. (b) A new random variable Z is given by Z = 2X + 1. Find the covariance between X and Z. (1pt) Find the covariance between Y and Z. (2pt)
Please do by hand. Thanks in advance. 8. Let X and Y be random variables with a bivariate normal distribution with parameters • px = 5 • Ox= 3 • My = 3 • Oy = 2 • p=.4 a) Find the expected value and variance of Z=4X-Y. b) Find the covariance of X and Z. c) Identify the distribution of Y. d) Identify the distribution of Y|X = 5.
Problem3 (15 points (a) (8 points) Let x, X, be a random sample from normal distribution NG, σ, . s are sample mean and sample variance. Consider the probabilities PC, μ) and PS? σ)-are they equal? (b) (7 points) Let X, , ,X, be a random sample from normal distribution Mo, σ, R, s are sample mean and sample variance. Let y.... is and independent sample from the same distribution. Y, s are corresponding sample mean and sample variance. Let...
Problem 1. (a) Let X be a Binomial random variable such that E(X) 4 and Var(x) 2. Find the parameters of X (b) Let X be a standard normal random variable. Write down one function f(t) so that the random variable Y-f(X) is normal with mean a and variance b.
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....
6. (15 points) Let X be an exponential random variable with mean 3. Answer the following: (a) Find the probability density function f(x). (b) Compute the probabilities P(2 < X < 6) and P(X 24). (c) Find the variance.