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Let Z and W be independent standard normal random
Chapter , Problem 6(choose chapter or problem)
Let Z and W be independent standard normal random variables. Let X and Y be defined by
X = σ1Z + μ1
Y = σ2[ρZ √1 - ρ2W] μ2
Where, σ1,σ2 > 0,–∞ < μ1, μ2 < ∞ and –1 < ρ < 1. Show that the joint probability density function of X and Y is bivariate normal and σ1 = σX,σ2 = σY,μ1 = μX,μ2 = μY and ρ = ρ(X,Y).
Questions & Answers
QUESTION:
Let Z and W be independent standard normal random variables. Let X and Y be defined by
X = σ1Z + μ1
Y = σ2[ρZ √1 - ρ2W] μ2
Where, σ1,σ2 > 0,–∞ < μ1, μ2 < ∞ and –1 < ρ < 1. Show that the joint probability density function of X and Y is bivariate normal and σ1 = σX,σ2 = σY,μ1 = μX,μ2 = μY and ρ = ρ(X,Y).
ANSWER:
Step 1 of 4
Let 
Here, 
To prove that the joint probability density function of
Since 
Now, 
It is a linear transformation of 