Problem # 7. X is uniform in the interval (−c, c). Find P[|X| ≤ σ 2 X].
Let Y = X2 where X is the random variable of Problem 7. Find the mean E[Y ] and variance σ 2 Y .
Problem # 7. X is uniform in the interval (−c, c). Find P[|X| ≤ σ 2...
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]
Let Y-ar+b (a) Find the mean and variance of Y in terms of the mean and variance of X b) Evaluate the mean and variance ofY if Xhas the following PDF: (a)-ele (c) Evaluate the mean and variance of Y if Xis the Gaussian random variable with mean 0 and variance d) Evaluate the mean and variance of Yif X-bcos 2U) where U is a uniform random variable in of 1 the unit interval. Let Y-ar+b (a) Find the mean...
Problem1 Let Y=aX + b . (a) Find the mean and variance of Y in terms of the mean and variance of X (b) Evaluate the mean and variance ofYifXhas the following PDF (c) Evaluate the mean and variance of Y if Xis the Gaussian random variable with mean 0 and variance of 1 d) Evaluate the mean and variance of Yif X bcos(2RU) where U is a uniform random variable in the unit interval. Problem1 Let Y=aX + b...
A Gaussian random variable X has mean 2 and variance 4 a) Find P(X < 3). (b) Find P(1 < X < 3) (c) Find P({X > 4}|{X > 3}) (d) Let Y = X^2 . Find E[Y].
Problem 2. Rice, Problem 7, pg. 314 (Extended)] Suppose that X1,..., Xn iid Geometric(p). a) Find the method of moments estimator for p. (b) Find the maximum likelihood estimator for p. (c) Find the asymptotic variance of the MLE (d) Suppose that p has a uniform prior distribution on the interval [0, 1]. What is the posterior distribution of p? For part (e), assume that we obtained a random sample of size 4 with L^^^xi-.4 (e) What is the posterior...
Let X be a normal random variable with mean 0 and variance σ^2. Find the density for |X|.
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...
2. Let X be a Bernoulli random variable with probability of X -1 being a. a) Write down the probability mass function p(X) of X in terms of a. Mark the range of a (b) Find the mean value mx(a) EX] of X, as a function of a (c) Find the variance σ剤a) IX-mx)2) of X, as a function of a. (d) Consider another random variable Y as a function of X: Y = g(X) =-log p(X) where the binary...
Problem5 Let x, ,x, be a random sample from normal population Na, σ Find method of moments estimator of σ: is it unbiased? Problem6 Random variable X has density f(x)-ax+ Bx' in the interval (0.1) and 0 elsewhere. Given that EX (a) find α, β, () find P Xx-o.s 0.09 (6) Let you have sample of size 25, with sample mean R.Estimate the probability R>0.8).Formulate the assumptions Problem5 Let x, ,x, be a random sample from normal population Na, σ...
2. Assume that the pdf of the random variable x is uniform in the interval (10, 12) and y = x^3. (a) Find fy (y). (b) Find E{y}.