Witam. Mam problem z takim zadankiem.
Niech, X i Y będą zmiennymi losowymi takimi, że X, Y ~ N(m, o), wsp. korelacji (X,Y)=1.
Obliczyć wsp. korelacji (U,V), gdy \(\displaystyle{ U=aX +bY}\), \(\displaystyle{ V=aX-bY}\), \(\displaystyle{ a^{2} + b^{2} >0}\).
Nie wiem jak policzyć kowariancje cov(U,V) bo mam cov(U,V)=E(UV)-E(U)E(V), E(U) policzę, E(V) też, ale nie wiem jak policzyć E(UV) :/.
Rozkład normalny
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bstq
- Użytkownik
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Rozkład normalny
\(\displaystyle{ \varrho_{XY}=1=\frac{Cov\left(X,Y\right)}{\sqrt{VarX\cdot VarY}}=\frac{Cov\left(X,Y\right)}{\sqrt{\sigma^{2}\cdot\sigma^{2}}}=\frac{Cov\left(X,Y\right)}{\sigma^{2}}\Rightarrow Cov\left(X,Y\right)=\sigma^{2}}\)
\(\displaystyle{ Cov\left(U,V\right)=Cov\left(aX+bY,aX-bY\right)=Cov\left(aX,aX-bY\right)+Cov\left(bY,aX-bY\right)=Cov\left(aX,aX\right)+Cov\left(aX,-bY\right)+Cov\left(bY,aX\right)+Cov\left(bY,-bY\right)=a^{2}Cov\left(X,X\right)-abCov\left(X,Y\right)+abCov\left(Y,X\right)-b^{2}Cov\left(Y,Y\right)=a^{2}Cov\left(X,X\right)-b^{2}Cov\left(Y,Y\right)=a^{2}VarX-b^{2}VarY=\left(a^{2}-b^{2}\right)\cdot\sigma^{2}}\)
\(\displaystyle{ VarU=Var\left(aX+bY\right)=Var\left(aX\right)+Var\left(bY\right)+2Cov\left(aX,bY\right)=a^{2}\cdot Var\left(X\right)+b^{2}\cdot Var\left(Y\right)+2abCov\left(X,Y\right)=\left(a^{2}+b^{2}\right)\sigma^{2}+2\cdot ab\cdot\sigma^{2}=\left(a+b\right)^{2}\sigma^{2}}\)
\(\displaystyle{ VarV=Var\left(aX-bY\right)=Var\left(aX\right)+Var\left(bY\right)-2Cov\left(aX,bY\right)=a^{2}\cdot Var\left(X\right)+b^{2}\cdot Var\left(Y\right)-2abCov\left(X,Y\right)=\left(a^{2}+b^{2}\right)\sigma^{2}-2\cdot ab\cdot\sigma^{2}=\left(a-b\right)^{2}\sigma^{2}}\)
\(\displaystyle{ \varrho_{UV}=\frac{Cov\left(U,V\right)}{\sqrt{VarU\cdot VarV}}=\frac{\left(a^{2}-b^{2}\right)\cdot\sigma^{2}}{\sqrt{\left(a+b\right)^{2}\sigma^{2}\left(a-b\right)^{2}\sigma^{2}}}=\frac{\left(a^{2}-b^{2}\right)\cdot\sigma^{2}}{\sqrt{\left(a^{2}-b^{2}\right)^{2}\left(\sigma^{2}\right)^{2}}}=\frac{\left(a^{2}-b^{2}\right)\cdot\sigma^{2}}{\left|a^{2}-b^{2}\right|\cdot\sigma^{2}}=\begin{cases}
-1 & a^{2}-b^{2}<0\\
1 & a^{2}-b^{2}>0\end{cases}}\)
\(\displaystyle{ Cov\left(U,V\right)=Cov\left(aX+bY,aX-bY\right)=Cov\left(aX,aX-bY\right)+Cov\left(bY,aX-bY\right)=Cov\left(aX,aX\right)+Cov\left(aX,-bY\right)+Cov\left(bY,aX\right)+Cov\left(bY,-bY\right)=a^{2}Cov\left(X,X\right)-abCov\left(X,Y\right)+abCov\left(Y,X\right)-b^{2}Cov\left(Y,Y\right)=a^{2}Cov\left(X,X\right)-b^{2}Cov\left(Y,Y\right)=a^{2}VarX-b^{2}VarY=\left(a^{2}-b^{2}\right)\cdot\sigma^{2}}\)
\(\displaystyle{ VarU=Var\left(aX+bY\right)=Var\left(aX\right)+Var\left(bY\right)+2Cov\left(aX,bY\right)=a^{2}\cdot Var\left(X\right)+b^{2}\cdot Var\left(Y\right)+2abCov\left(X,Y\right)=\left(a^{2}+b^{2}\right)\sigma^{2}+2\cdot ab\cdot\sigma^{2}=\left(a+b\right)^{2}\sigma^{2}}\)
\(\displaystyle{ VarV=Var\left(aX-bY\right)=Var\left(aX\right)+Var\left(bY\right)-2Cov\left(aX,bY\right)=a^{2}\cdot Var\left(X\right)+b^{2}\cdot Var\left(Y\right)-2abCov\left(X,Y\right)=\left(a^{2}+b^{2}\right)\sigma^{2}-2\cdot ab\cdot\sigma^{2}=\left(a-b\right)^{2}\sigma^{2}}\)
\(\displaystyle{ \varrho_{UV}=\frac{Cov\left(U,V\right)}{\sqrt{VarU\cdot VarV}}=\frac{\left(a^{2}-b^{2}\right)\cdot\sigma^{2}}{\sqrt{\left(a+b\right)^{2}\sigma^{2}\left(a-b\right)^{2}\sigma^{2}}}=\frac{\left(a^{2}-b^{2}\right)\cdot\sigma^{2}}{\sqrt{\left(a^{2}-b^{2}\right)^{2}\left(\sigma^{2}\right)^{2}}}=\frac{\left(a^{2}-b^{2}\right)\cdot\sigma^{2}}{\left|a^{2}-b^{2}\right|\cdot\sigma^{2}}=\begin{cases}
-1 & a^{2}-b^{2}<0\\
1 & a^{2}-b^{2}>0\end{cases}}\)