2. (a) Suppose X,~ N(0,6), for i = 1, 2, 3, 4. Assume all these random variables areindependent. Derive the value of k in each of the following.i. P(X₁ + 8X₂ < k) = 0.2119.ii. P= 0.99.(2x7²<*) =iii. P· P (X₁ > (k(X² + X² + X²))¹/²) = = 0.05.

Given that Xi~N0, 6 i=1,2,3,4 and Xi's are independent

Answered: Suppose X₁~ N(0, 6), for i = 1, 2, 3,…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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3. Consider two random variables X₁ and X2 whose joint pdf is given by for x₁0, x2 > 0, x₁ + x2 < 2, { Find the pdf of Y = X₂ - X₁. f(x1, x₂) = NIT otherwise.
Q7. Let X; (i = 1,.,n) be a random sample from the N(u, o²) population with unknown parameters u and o?. Then we know that T, = E-1(X - X)²/(n – 1) is an unbiased estimator -xi-1 . Now consider two other estimators T, and T3 for o? where of a? and further, (n-1)T T2 = E(X – X)*/n and T3 = E(X – X)²/n + 1). %3D What is the variance of T,? Find out the variances and biases of T2 and T3 and thus obtain their m.s.e.'s. Show that among the three estimators T,, T2 and T3 the one with minimum m.s.e. is T3.
Suppose that X is a random variable which takes values in the set {1, 2, 3}. If P(X = 1) = 0.2 and E(X) = 2, determine the fourth moment E(X4). Answer:
None
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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.
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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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Suppose X₁~ N(0, 6), for i = 1, 2, 3, 4. Assume all these random variables are independent. Derive the value of k in each of the following. i. P(X₁ +8X₂ < k) = 0.2119. ii. P ( 2 X ² < k) = 0 = 0.99. iii. P(X₁> (k(X2 + X3 + X))¹/2) = 0.05.