Q 6.1. Suppose Z = (Z₁, Z2, Z3) is a standard multi-variate Gaussian random variable i.e., fori ≤ 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 onerandom variables, (1)-(4), on the right. Pair up according to "equal in distribution" and explainbriefly your reasoning.(a)X₁X2=(√2)X₁(b) (x₂)-(2₂2₂)=Z₁X₁ 3/√√2√2X₂=IV2) (21)01/√√21/√2 (Z₁(c)31(¹) (X) - (11) (2)(d1(1) (2)-()) (²)=Y₂(2) (Y/₁2)(12) = (1/✓/²2²(3)1/2) (²)1/√√2 1/√√21/√√2 -1/√2,2 2 √2(X) - (² : ¹3) (2)10(4) (1/12)(12) - (² 1) (²2)=

Given that be a standard multi-variate Gaussian random variable.For are i.i.d. random variables.

Answered: Q 6.1. Suppose Z = (Z1, Z2, Z3) is a…

A First Course in Probability (10th Edition)

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Chapter1: Combinatorial Analysis

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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Let X1, X2, X3, and X4 be a random sample of observations from a population with mean μ and variance σ2. Consider the following estimator of μ: �"!= 0.15 X1 + 0.35 X2 + 0.20 X3 + 0.30 X4.What is the variance of � "!
1.
A docs.google.com/forms/d/e (1) Let X be a random variable with p.d.f. fx) =- , then O.w (b) k- (0) k = (a) k=1 (d) k-5 (e) None of these a is the correct answer b is the correct answer c is the correct answer d is the correct answer e is the correct answer (2) Let X be a random variable with a function x= 0,1, 2,3 4 x! (3-x)! f(x) = , then p(x<1) is 0.w (b) (d) (e) None of these 27 (a) 27 (c) 27 a is the correct answer b is the correct answer c is the correct answer d is the correct answer e is the correct answer
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5. Suppose there are 2 populations, each with mean ₁ and 4₂ and standard deviation of ₁ and 2 respectively. The interest is to measure a random variable X by taking a random sample of size n from the first population. In a similar manner, another random variable Y is measured by independently taking a random sample of size m from the second population. Assuming n and m are sufficiently large, then demonstrate the following properties: a. X~N (μ₁, 1) b. 8~ N(H₂. /) c. (8-8)-№ (1₁ - 1₂ + 15t m
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Let Y1, Y2, Y3 denote independent exponential random variables with respective rates A; for i e {1,2, 3}. Answer the following questions; (a) Compute E[Y1 +Y2 +Y3]Y1 > 1, Y2 > 2, Y3 > 3]. Your answer should be an expression using A;'s. (b) Compute the probability that Yı is the smallest among three random variables. Hint: P(X < Y) = S" Px,y (x, y)dædy where Px,Y (x, y) is the joint density function.
for the first part, could u tell me why the lower bound is 0 rather than x? We can see from the question that 0<x<y<1, if u set the lower bound=0, x might be bigger than y
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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. (a) (X₂) (¹) (X) = (2) 1/√2 (Ⓒ) (x₂) - (3/1² ¹1/1²) (2) (c = 2 0 = 2 Z₁ 1) (²) 3 (4¹) (X) = (²¹) (1) (2) (3) Y₁ 1 (29) - ()) (2) = Y₂ 1 Y₁ (121) = (1/² √₂) (2) (1/√2 1/√2 /2 −1/√√2, Y₂ X₁ X₂ = 22 1 1 2 0 Y₁ 2 1 (4) (2)) = (²) (²) 1 1 Z₁ Z₂ Z3

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