2. Assume the random variable y has the continuous uniform distribution defined on the interval a...
5. A continuous random variable X follows a uniform distribution over the interval [0, 8]. (a) Find P(X> 3). (b) Instead of following a uniform distribution, suppose that X assumes values in the interval [0, 8) according to the probability density function pictured to the right. What is h the value of h? Find P(x > 3). HINT: The area of a triangle is base x height. 2 0 0
3. (5 marks) Let U be a random variable which has the continuous uniform distribution on the interval I-1, 1]. Recall that this means the density function fu satisfies for(z-a: a.crwise. 1 u(z), -1ss1, a) Find thc cxpccted valuc and the variancc of U. We now consider estimators for the expected value of U which use a sample of size 2 Let Xi and X2 be independent random variables with the same distribution as U. Let X = (X1 +...
(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) =
Problem 3: Assume the continuous random variable X follows the uniform[0,1] distribution, and define another random variable Y- In () 1-X a) Determine the CDF of Y. Hint: start by writing P(Y y), then show that P(Y y) = P(X s g(v)), where g(y) is a function that you need to determine. b) Determine the PDF of Y.
Assume the continuous random variable X follows the uniform [0,1] distribution, and define another random variable We were unable to transcribe this imagea) Determine the CDF of Y. Hint: start by writing P(Y ), then show that P(Y y) = P(X s g(v)), where g(y) is a function that you need to determine. b) Determine the PDF of Y.
.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
2. Let R be a continuous random variable with the following conditional distribution Rlo~N(0, o2) where o is a discrete valued random variable with the following distribution 1 P(o 1) P(o 100) =. 2 Compute the mean, variance and kurtosis of the random variable R. Hint: use the double expectation formula 2. Let R be a continuous random variable with the following conditional distribution Rlo~N(0, o2) where o is a discrete valued random variable with the following distribution 1 P(o...
(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) =
Assume that the density function for a continuous random variable, Y, is defined as fY(y) = 9y. exp(-3y) for (y>0) and f'(y) = 0 elsewhere. Given Y = y, the conditional C.D.F. for X is FX\Y (x\Y) =P[X 5 X Y = y) = 1 – exp(-x •y) for (x > 0). Questions below are related to the marginal distribution for X. 1. Derive the density, f* (x). 2. Evaluate the expectation, E(X)
Assume a continuous random variable X has a continuous uniform distribution between 35 and 48. Find P(X<38)