-wa exp{-(20 )2}, where The Normal(μ,02) distribution has density f(x) -oo < μ < oo and...
2. A randon sample XI, X. is drawn frotn Normal(μ, σ2), where-oo < μ < oo and 0 < σ2 < x. To test the null hypothesis Ho : σ2-1 against the alternative H1: σ2 > 1, we have designed the following test Reject Ho if S>k where S2 = "LE:-1(x,-X)2, k ís a constant. Noticed that (n-1) distribution with degree of freedom 1 has a (a) Determine k so that the test will have size a. (b) Use k...
4. The moment generating function of the normal distribution with parameters μ and σ2 is (t) exp ( μ1+ σ2t2 ) for -oo < t oo. Show that E X)-ψ(0)-μ and Var(X)-ψ"(0)-[ty(0)12-σ2. 5. Suppose that X1, X2, and X3 are independent random variables such that E[X]0 and ElX 1 for i-12,3. Find the value of E[LX? (2X1 X3)2] 6. Suppose that X and Y are random variables such that Var(X)-Var(Y)-2 and Cov(X, Y)- 1. Find the value of Var(3X -...
In question 5, f(x) = λ*exp(-λx), for x greater or equal to 0, and zero otherwise. 9. Let X have an exponential distribution with λ = 1 (see Question 5), and let Y = log(X). Find the probability density function of Y. Where is the density non-zero? Note that in this course, log refers to the log base e, or natural log, often symbolized In. The distribution of Y is called the (standard) Gumbel, or extreme value distribution. 2
(2) Let X be a locally compact Hausdorff space, and let μ be a regular Borel measure on X such that μ(X) = +oo. Show that there is a non-negative function f CO(X) such that Jfdlı-+oo. Idea. Construct a sequence {K f-Σ001 nzfn, n} of disjoint compact sets K n with μ(An) > n and set where fn E Co(X) with XKn S f 31 く! (2) Let X be a locally compact Hausdorff space, and let μ be a...
Problem 5. Suppose that the continuous random variable X has the distribution fx(z),-oo < x < oo, which is symmetric about the value x-0. Evaluate the integral: Fx (t)dt -k where Fx(t) is the CDF for X, and k is a non-negative real number.
5. The Exponential(A) distribution has density f(x) = for x<0' where λ > 0 (a) Show/of(x) dr-1. (b) Find F(x). Of course there is a separate answer for x 2 0 and x <0 (c Let X have an exponential density with parameter λ > 0 Prove the 'Inemoryless" property: P(X > t + s|X > s) = P(X > t) for t > 0 and s > 0. For example, the probability that the conversation lasts at least t...
Let fy(x, μ, σ) stand for the probability distribution function (PDF) for the normal distribution with parameters μ and σ. Let X be a random variable with a PDF defined as follows: where t is a fixed constant between O and 1. What is E[XI? None of these
2. (10pts) Let X1, X2, , X20 be an i.i.d. sannple from a Normal distribution with mean μ and variance σ2, ie., Xi, X2, . . . , X20 ~ N(μ, σ2), with the density function Also let 20 20 10 20 -20 19 i-1 ー1 (a) (5pts) What are the distributions of Xi - X2 and (X1 - X2)2 respectively? Why? (b) (5pts) what are the distributions of Y20( and 201 ? Why? (X-μ)2 2. (10pts) Let X1, X2,...
X follows normal distribution N (μ, σ2) with pdf f and cdf F. If max, f (x)-0.997356 and F (-1) + F (7-1, determine P(X s 0)
X follows normal distribution N (μ, σ2) with pdf f and cdf F. if max, f (z) = 0.997356 and F (-1) + F (7)-1, determine .4 .4