D. Assume that X,,X2, ...,X, is a random sample from a r'(1,0) distribution1.1. The moment generating function of the random variable =2-1xi is given by2 r9M."22 () = (1– 2t)- 9.TrueFalse2. The random variable E-, x; has a X2-distribution with 9 degrees of freedom.219i3D1TrueO False2.

Answered: D. Assume that X1,X2, ..., X9 is a…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

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B4. Let X₁,... Xn ~ N(μ, o2) be independent random variables. 2 (a) From lectures we know (X=X) ²³. X₂ (b) Let n i=1 ~ x²(v). What is the value of v? n 1 *³ =, ²-₁, [(x₁ - x) ². Σ(x s n 1 i=1 Determine Var(s), that is, the variance of the sample variance. (c) Assume now that we have observed data ₁,...,n ER with sample variance s². Use the result from part (a) to find numbers an and b, such that [ans, bns] is an exact 95%-confidence interval for o².
11. Let (y1, Y2 ... Yn) be independent random sample from the uniform distribution on [0, 1]. (a) show that Z = – In Y; has exponential distribution with parameter 1. (b) Hence or otherwise, show that -2 In Y; xản
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10. Let x be a binomial random variable with n = 150 and p = 0.4. Use the normal approximation to find the following. a) P(48 69) P(X>69) = 1- P(X = 69.5) = 1−ø(- c) P(X ≤ 70) 70.5-60. 69.5–60. 6 P(X <=70) = P(X = 70.5) = (- -) = = 0.9599 -) = 0.0571
Q4: Let i denote the mean of a random sample of size 25 from a gamma type distribution with a = 4 and p > 0. Then 1. 1. E(x) = 48 and Var(x) = 48² O True O False 2. I~N (4B, 4B²) 25 2. O True O False
2) Let X₁, X2, ..., Xn be a random sample from the pdf f(x;0) = 0xª−¹, 0≤x≤ 1,0 <0 < ∞. Find the MLE of 0 and show that its variance approaches 0 as n approaches ∞o.
5. Let X₁ and X₂ be iid Poisson() random variables with pmf p(x) = H'e exp{p(e-1)), for tER. Let Y=X₁+X₂. a. b. What is the distribution of Y? What are E[Y] and Var[Y]? for x 0,1,2,..., and mgf 9 X!
Q 6.1. Suppose Z = (Z₁, Z2, Z3) is a standard multi-variate Gaussian random variable i.e., for i ≤ 3, Zi~ N(0, 1) are i.i.d. random variables. Each of the random variables, (a)–(d), on the left is equal in distribution to exactly one random variables, (1)–(4), on the right. Pair up according to "equal in distribution" and explain briefly your reasoning. X₁ (a) X₂ (b) (x₂) - (2) (c) (x²) - (¹/1² 1/√²) (2₁) (3/√2 = √2 = (V2) Z₁ (d) (x) - (1) (²) X2 Y₁ (9)-(-) (2) (1=1) Y₂ (1) Y₁ (²) (4) - (1/² (3) (1/√√2 1/√2-1/√2) (x₁) = (²₁ ² ✓/³²) (3) 2 2 √2 1 Z3 Y₁ → ()-()) (4) = Y₂ 1/2) (2) 1) (2)

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D. Assume that X1,X2, ..., X9 is a random sample from a r(1,0) distribution - The moment generating function of the random variable £-,x; is given by M. "a2 (t) = (1– 2t)- 9.