3. (25 points) Let be a uniform random variable on the interval (0,2m]. Define a cos...
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]
Assume random variable ? is uniformly distributed in the interval (−?/2 ,?⁄ 2]. Define the random variable ?=tan (?), where tan (∙) denotes the tangent function. Note that the derivative of tan (?) is 1/(cos (?)2) . a) Find the PDF of ?. b) Find the mean of ? .Define the random variable ?=1/?. c) Find the PDF of ?. Assume random variable X is uniformly distributed in the interval (-1/2, 1/2). Define the random variable Y = tan(X), where...
.A Uniform random variable is defined on the interval (8,0+1). Let YY, be a random sample taken from this distribution and Y is an estimator of θ. (a) Compute the bias of Y (b) Find a function of that is an unbiased estimator of θ. (c) Find MSE(Y
1) 2) 3) 4) 5) Suppose that X is a uniform random variable on the interval (0, 1) and let Y = 1/X. a. Give the smallest interval in which Y is guaranteed to be. Enter -Inf or Inf for – or o. Interval:( b. Compute the probability density function of Y on this interval. fy(y) = Suppose that X ~ Bin(4, 1/3). Find the probability mass function of Y = (X – 2)2. a. List all possible values that...
(20 pts) Let U be a random variable following a uniform distribution on the interval [0 Let X=2U + 1 (a) Is X a random variable? Why or why not? (b) Calculate E[X] analytically
(5 pts) Let U be a random variable following a uniform distribution on the interval [0, 1]. Let X=2U + 1 Calculate analytically the variance of X. (HINT : Elg(z)- g(z)f(x)dr, and the pdf. 0 < z < 1 0 o.t.w. f(x) of a uniform distribution is f(x) =
complete using R 3. Let X ~ Unif(1,2) be a uniform random variable on the interval (1,2). (a) What is the exact value of the mean of X? (b) Compute or estimate the standard deviation of X. (c) Estimate the expected value E[1/X] accurately to two decimal places.
3. (Bpoints) Let X, Y and Z be independent uniform random variables on the interval (0, 2), Let W min(X, y.z a) Find pdf of W Find E(1-11 b) 3. (Bpoints) Let X, Y and Z be independent uniform random variables on the interval (0, 2), Let W min(X, y.z a) Find pdf of W Find E(1-11 b)
(5 pts) Let U be a random variable following a uniform distribution on the interval [0,1 Let Calculate analytically the variance of X. (HINT: E g(x)f(x)dx, and the p.d.f. 10SzSI 0 o.t.w. f(x) of a uniform distribution is f(x) =
Let Θ be a continuous random variable uniformly distributed on [0,2 Let X = cose and Y sin e. Show that, for this X and Y, X and Y are uncorrelated but not independent. (Hint: As part of the solution, you will need to find E[X], E[Y] and E|XY]. This should be pretty easy; if you find yourself trying to find fx(x) or fy (v), you are doing this the (very) hard way.) Let Θ be a continuous random variable...