Question

(2) Dem que dos variables al son incorclades sieupre que seavi indeper. \[ \begin{array}{l} x, y \text { incors. si } P(x, y)=0=\frac{\operatorname{cov}(x, y)}{D T(x) D T(Y)} \Leftrightarrow \operatorname{cov}(x, y)=0 \Leftrightarrow \\ \Leftrightarrow E((X-E(X))(Y-E(y)))=0 \Leftrightarrow E(X Y-X E(Y)-Y E(X)+E(X) E(Y))=0 \\ \operatorname{cov}^{\prime \prime}(x, y) \\ \Leftrightarrow E(x y)-E(y) E(x)-E(x) E E(y)+E(x) E(y)=0 \quad \text { QE0. } \\ \end{array} \]

          (2) Dem que dos variables al son incorclades sieupre que seavi indeper.
\[
\begin{array}{l}
x, y \text { incors. si } P(x, y)=0=\frac{\operatorname{cov}(x, y)}{D T(x) D T(Y)} \Leftrightarrow \operatorname{cov}(x, y)=0 \Leftrightarrow \\
\Leftrightarrow E((X-E(X))(Y-E(y)))=0 \Leftrightarrow E(X Y-X E(Y)-Y E(X)+E(X) E(Y))=0 \\
\operatorname{cov}^{\prime \prime}(x, y) \\
\Leftrightarrow E(x y)-E(y) E(x)-E(x) E E(y)+E(x) E(y)=0 \quad \text { QE0. } \\
\end{array}
\]

(2) Dem que dos variables al son incorclades sieupre que seavi indeper.

x, y incors. si P(x, y)=0=(cov(x, y))/(D T(x) D T(Y))⇔cov(x, y)=0 ⇔
⇔ E((X-E(X))(Y-E(y)))=0 ⇔ E(X Y-X E(Y)-Y E(X)+E(X) E(Y))=0
cov^''(x, y)
⇔ E(x y)-E(y) E(x)-E(x) E E(y)+E(x) E(y)=0 QE0.

Added by Ivan C.

Differential Equations

E. RUKMANGADACHARI 2012 Edition

Chapter 2

Instant Answer

Solved by Expert Narayan Hari

Step 1

The question asks to demonstrate that two variables are uncorrelated if and only if their covariance is zero. The formula given is \(P(x, y) = 0 = \frac{\operatorname{cov}(x, y)}{DT(x) DT(Y)}\), which simplifies to \(\operatorname{cov}(x, y) = 0\). Show more…

Show all steps

Thanks for your feedback!

(2) Dem que dos variables al son incorclades sieupre que seavi indeper. \[ \begin{array}{l} x, y \text { incors. si } P(x, y)=0=\frac{\operatorname{cov}(x, y)}{D T(x) D T(Y)} \Leftrightarrow \operatorname{cov}(x, y)=0 \Leftrightarrow \\ \Leftrightarrow E((X-E(X))(Y-E(y)))=0 \Leftrightarrow E(X Y-X E(Y)-Y E(X)+E(X) E(Y))=0 \\ \operatorname{cov}^{\prime \prime}(x, y) \\ \Leftrightarrow E(x y)-E(y) E(x)-E(x) E E(y)+E(x) E(y)=0 \quad \text { QE0. } \\ \end{array} \]