X ∼ Poisson(θ) and independently, Y ∼ binomial(2, θ). Z = XY. (c) Define W = Z−(1+2θ−θ2)X. Are Z and W independent? Justify our answer. E[Z] = 2θ2, Var(Z) = 2θ2(1+2θ−θ2), Cov(X, Z) = 2θ2, Cov(X, Z) = 2θ2−2θ3 are known.
X ∼ Poisson(θ) and independently, Y ∼ binomial(2, θ). Z = XY. (c) Define W = Z−(1+2θ−θ2)X. Are Z and W independent? Justify our answer. E[Z] = 2θ2, Var(Z) = 2θ2(1+2θ−θ2), Cov(X, Z) = 2θ2, Cov(X, Z) = 2θ2−2θ3 are known.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.2: Systems Of Linear Equations In Two Variables
Problem 50E
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X ∼ Poisson(θ) and independently, Y ∼ binomial(2, θ). Z = XY.
(c) Define W = Z−(1+2θ−θ2)X. Are Z and W independent? Justify our answer.
E[Z] = 2θ2, Var(Z) = 2θ2(1+2θ−θ2), Cov(X, Z) = 2θ2, Cov(X, Z) = 2θ2−2θ3 are known.
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