14. Let X₁, X2,... be i.i.d. r.v.'s with finite expectation 0 and finite variance 1, and define the r.v.'sYn and Zn byYn = n√n;in XiX?'n(a) Prove that Σ=1X² ²1 as n → ∞.nd(b) Show that YnZ~ N (0, 1) as n →∞.(c) Show that ZnZ ~ N(0, 1) as n →∞.Zn= nΣi=1 Xi(Σ=1X3)1/2 °

a) Convergence in Probability: limn→∞PXn-X≥ε=0, for all ε>0 From the given information,…

Answered: 4. Let X₁, X2,... be i.i.d. r.v.'s with…

A First Course in Probability (10th Edition)

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Publisher:Sheldon Ross

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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6. Suppose that X e N(0, 1) and Y E Exp(1) are independent random variables. Prove that XV2Y has a standard Laplace distribution.
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Suppose there are three possible states of nature, and the class-conditional PDFS are the Cauchy distributions 1 p(rw;) = 1 i = 1,2,3. (1) 2 1+ bị Let ai = -2, a2 = 0, az = 2, bị = b2 = 1, b3 = 0.5. Let P(wi) = P(w2) = 0.4, P(w3) = 0.2. (a) Find the decision boundaries and the decision regions for the Bayesian decision (b) Calculate the probability of error for the classification done according to this
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)
4.Suppose that X is a random variable for which E(X)=µ and Var(X)=o². Show that E(X(X-1)]=µ(µ-1)+o?
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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4. Let X₁, X2,... be i.i.d. r.v.'s with finite expectation 0 and finite variance 1, and define the r.v.'s Yn and Zn by τη τ 1 Yn = nynΣ 1 X xar Σ=1 D Σ=1 X (Σ=1X?)1/ , Zn = n· 1/2