Let Ybe a normal random variable with parameters (1,a2). In other words, its mean is 1 while its variance a2 is unknown...
Problem 1: confidence interval for a variance parameter for a normal distribution Let Ybe a normal random variable with mean μand variance σ2. Assume that μis known but σ2is unknown. Show that ((Y-μ)/σ)2is a pivotal quantity. Use this pivotal quantity to derive a 1-α confidence interval for σ2. (The answer should be left in terms of critical values for the appropriate distribution.)
3. Suppose that the random variable X is an observation from a normal distribution with unknown mean μ and variance σ (a) 95% confidence interval for μ. (b) 95% upper confidence limit for μ. (c) 95% lower confidence limit for μ. 1 . Find a
6. Let Xi 1,... ,Xn be a random sample from a normal distribution with mean u and variance ơ2 which are both unknown. (a) Given observations xi, ,Xn, one would like to obtain a (1-a) x 100% one-sided confidence interval for u as a form of L E (-00, u) the expression of u for any a and n. (b) Based on part (a), use the duality between confidence interval and hypothesis testing problem, find a critical region of size...
8.40 stion 4 (6 pt) (Ex. 8.40 on page 409 is modified): Suppose that random variable Y is an observation from a normal distribution with unknown mean u and variance l Find and verify a pivotal quantity that you can use to derive confidence limits for the mean u. Find a 95% lower confidence limit for. a. b. 8.40 Suppose that the random variable Yis an observation from a normal distribution with unknown mean μ and variance 1 . Find...
Problem 1. Let X be a normal random variable with mean 0 and variance 1 and let Y be uniform(0.1) with X and Y being independent. Let U-X + Y and V = X-Y. For this problem recall the density for a normal random variable is 2πσ2 (a) Find the joint distribution of U and V (b) Find the marginal distributions of U and V (c) Find Cov(U, V).
Assume X is a normal random variable with mean 20 and variance 16, and Y is a Gamma random variable with parameters 5 and 2. In addition X and Y are independent. Construct a box with length L = [X], width W = 2|X|, and height H = Y. Let V be the volume of the box. Calculate the expected value E[V]?
Let X1 be a normal random variable with mean 2 and variance 3, and let X2 be a normal random variable with mean 1 and variance 4. Assume that X1 and X2 are independent. What is the distribution of the linear combination Y = 2X1 + 3X2?
Let X be a normal random variable with mean 0 and variance 0.5 and Y be exponentially distributed with mean 1. Suppose X and Y are independent. Find P(Y>X2 ).
Let X be a zero-mean normal distributed random variable with variance of 2. Let Y gx), where 4 -2542-1 120 0, Find the CDF and PDF of the random variable Y.
Let X be a zero-mean normal distributed random variable with variance of 2. Let Y gx), where 4 -2542-1 120 0, Find the CDF and PDF of the random variable Y.