Determine the correlation coefficient of the random variables X and Y if var(X) = 4, var(Y ) = 2, and var(X + 2Y ) = 15.

Hana Buck

Hana Buck

Answered question

2022-09-23

Determine the correlation coefficient of the random variables X and Y if var(X) = 4, var(Y ) = 2, and var(X + 2Y ) = 15.

Answer & Explanation

Nancy Phillips

Nancy Phillips

Beginner2022-09-24Added 12 answers

The correlation coefficient of X and Y is
ρ X , Y = C o v ( X , Y ) σ X σ Y .
By the properties of variance
V a r ( X + 2 Y ) = V a r ( X ) + 2 2 V a r ( Y ) + 2 1 2 C o v ( X , Y )
= σ X 2 + 4 σ Y 2 + 4 ρ σ X σ Y ,
and from the fact that
Var(X+2Y)=15
it follows that
σ X 2 + 4 σ Y 2 + 4 ρ σ X σ Y = 15
4 + 4 2 + 4 ρ 4 2 = 15
which implies that
ρ = 0.27
Result:
ρ = 0.27

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