TU Wien:Statistik und Wahrscheinlichkeitstheorie UE (Levajkovic)/Übungen 2023W/HW12.1

Correlation 1

Let $X$ and $Z$ be two uncorrelated random variables with $Var(X)=Var(Z)=1$, and let $Y=k\cdot X+{\sqrt {(1-k)^{2}}}\cdot Z,{\text{ for }}k\in [-1,1].$ Compute the correlation between $X$ and $Y$.

Note: $Y$ depends linearly on $X$ and linearly on $Z$.

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Lösungsvorschlag von Lessi

--Lessi 2024-02-07T13:04:11Z

Since $X$ and $Z$ are uncorrelated we know:

${\begin{aligned}\mathbb {C} orr(X,Z)={\frac {\mathbb {C} ov(X,Z)}{\sqrt {\mathbb {V} ar(X)\cdot \mathbb {V} ar(Z)}}}=0\implies \mathbb {C} ov(X,Z)=0\ {\text{ (1)}}\end{aligned}}$

Also because of that we can determine $\mathbb {V} ar(Y)$:

${\begin{aligned}\mathbb {V} ar(Y){\stackrel {(1)}{=}}\mathbb {V} ar(k\cdot X)+\mathbb {V} ar({\sqrt {1-k^{2}}}\cdot Z)=k^{2}\mathbb {V} ar(X)+(1-k^{2})\mathbb {V} ar(Z)=1\ {\text{ (2)}}\end{aligned}}$

Now we can find the correlation:

${\begin{aligned}\mathbb {C} orr(X,Y)&={\frac {\mathbb {C} ov(X,Y)}{\sqrt {\mathbb {V} ar(Y)}}}\\&{\stackrel {(2)}{=}}\mathbb {C} ov(X,k\cdot X+{\sqrt {1-k^{2}}}\cdot Z)\\&=\mathbb {C} ov(X,k\cdot X)+\mathbb {C} ov(X,{\sqrt {1-k^{2}}}\cdot Z)\\&=k\cdot \mathbb {C} ov(X,X)+{\sqrt {1-k^{2}}}\cdot \mathbb {C} ov(X,Z)\\&{\stackrel {(1)}{=}}k\end{aligned}}$