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5) Let (X₁, Y₁),,(X,Y) be a random sample from the bivariate distribution with the joint p.d.f. f(x,y)= e = (x+y), 0 < x,y<∞ with E(X) = E(Y) = k!, k=1,2,.... a) Let Dn. n Let X₂= EX/n, Y= Y/n. (30) i=1 = n n [(X, i=1 i=1 Y)²/(2n)]¹/2, show that D₁ → 1 in probability as n→∞o. (10) b) Find the approximate distribution g(x) = ln (1+ X) for large n. (10) c) Find the approximate distribution of R₁ = g(X, Yn) = X₂/(1+ Y₂) for large n. (10)

5) Let (X₁, Y₁),,(X,Y) be a random sample from the bivariate distribution with the joint p.d.f. f(x,y)= e = (x+y), 0 < x,y<∞ with E(X) = E(Y) = k!, k=1,2,.... a) Let Dn. n Let X₂= EX/n, Y= Y/n. (30) i=1 = n n [(X, i=1 i=1 Y)²/(2n)]¹/2, show that D₁ → 1 in probability as n→∞o. (10) b) Find the approximate distribution g(x) = ln (1+ X) for large n. (10) c) Find the approximate distribution of R₁ = g(X, Yn) = X₂/(1+ Y₂) for large n. (10)

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
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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5) Let (X₁,Y₁),..., (X,Y) be a random sample from the bivariate distribution with the joint p.d.f. - (x + y) f(x,y)= = e ), 0<x,y<∞ with E(X)=E(Y) = k!, k=1,2,.... a) Let Dn n Let X₂= [X;/n, Y₂= Y;/n. (30) i 1 = n i=1 n [Σ(x₂ - y₂)²/(2n)]¹/2, show that D₁ → 1 in probability as n→∞. (10) i=1 b) Find the approximate distribution g(x) = ln (1+ X₂ ) for large n. (10) c) Find the approximate distribution of R₁ = g(X₁, Y₂) = X₂/(1+ Y₂) for large n. (10)
Transcribed Image Text:5) Let (X₁,Y₁),..., (X,Y) be a random sample from the bivariate distribution with the joint p.d.f. - (x + y) f(x,y)= = e ), 0<x,y<∞ with E(X)=E(Y) = k!, k=1,2,.... a) Let Dn n Let X₂= [X;/n, Y₂= Y;/n. (30) i 1 = n i=1 n [Σ(x₂ - y₂)²/(2n)]¹/2, show that D₁ → 1 in probability as n→∞. (10) i=1 b) Find the approximate distribution g(x) = ln (1+ X₂ ) for large n. (10) c) Find the approximate distribution of R₁ = g(X₁, Y₂) = X₂/(1+ Y₂) for large n. (10)
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