91Ó°ÊÓ

Probability simply refers to the likelihood of something occurring. We may talk about the probabilities of particular outcomes—how likely they are—when we're unclear about the result of an event. Statistics is the study of occurrences guided by probability.

In the above example, we can see how to compute covariance and expectation. We are provided the following information.

\(\begin{aligned}{l}{\rm{Cov(X,Y) = - }}\frac{{\rm{2}}}{{{\rm{75}}}}\\{\rm{E(X) = E(Y) = }}\frac{{\rm{2}}}{{\rm{5}}}\end{aligned}\)

The

correlation coefficient

of \({\rm{X}}\) and \({\rm{Y}}\) is

\({{\rm{\rho }}_{{\rm{X,Y}}}}{\rm{ = }}\frac{{{\rm{Cov(X,Y)}}}}{{{{\rm{\sigma }}_{\rm{X}}}{\rm{ \times }}{{\rm{\sigma }}_{\rm{Y}}}}}\)

We need only to calculate variances of both random variables.

Remember that

\({\rm{V(X) = E}}\left( {{{\rm{X}}^{\rm{2}}}} \right){\rm{ - (E(X)}}{{\rm{)}}^{\rm{2}}}\)

The following is true since we have the pdf of both random variables (and the polf is the same).

\(\begin{aligned}{\rm{E}}\left( {{{\rm{X}}^{\rm{2}}}} \right){\rm{ = }}\int_{\rm{0}}^{\rm{1}} {{{\rm{x}}^{\rm{2}}}} {{\rm{f}}_{\rm{X}}}{\rm{(x)dx}}\\{\rm{ = }}\int_{\rm{0}}^{\rm{1}} {\rm{1}} {\rm{2}}{{\rm{x}}^{\rm{2}}}{\rm{x(1 - x}}{{\rm{)}}^{\rm{2}}}{\rm{dx}}\\{\rm{ = 12}}\int_{\rm{0}}^{\rm{1}} {{{\rm{x}}^{\rm{3}}}} \left( {{\rm{1 - 2x + }}{{\rm{x}}^{\rm{2}}}} \right){\rm{dx}}\\{\rm{ = 12 \times }}\left( {\left. {\frac{{{{\rm{x}}^{\rm{4}}}}}{{\rm{4}}}} \right|_{\rm{0}}^{\rm{1}}{\rm{ - }}\left. {{\rm{2}}\frac{{{{\rm{x}}^{\rm{5}}}}}{{\rm{5}}}} \right|_{\rm{0}}^{\rm{1}}{\rm{ + }}\left. {\frac{{{{\rm{x}}^{\rm{6}}}}}{{\rm{6}}}} \right|_{\rm{0}}^{\rm{1}}} \right)\\{\rm{ = 12 \times }}\left( {\frac{{\rm{1}}}{{\rm{4}}}{\rm{ - }}\frac{{\rm{2}}}{{\rm{5}}}{\rm{ + }}\frac{{\rm{1}}}{{\rm{6}}}} \right)\\{\rm{ = }}\frac{{\rm{1}}}{{\rm{5}}}{\rm{.}}\end{aligned}\)

Therefore, the variance is

\(\begin{aligned}{\rm{V(X) = E}}\left( {{{\rm{X}}^{\rm{2}}}} \right){\rm{ - (E(X)}}{{\rm{)}}^{\rm{2}}}\\{\rm{ = }}\frac{{\rm{1}}}{{\rm{5}}}{\rm{ - }}{\left( {\frac{{\rm{2}}}{{\rm{5}}}} \right)^{\rm{2}}}\\{\rm{ = }}\frac{{\rm{1}}}{{{\rm{25}}}}\end{aligned}\)

The same way we compute the variance for\({\rm{Y}}\), which is the same before the marginal pdf's are the same, we compute the variance for\({\rm{Z}}\).

\({\rm{V(Y) = }}\frac{{\rm{1}}}{{{\rm{25}}}}\)

Finally, the correlation coefficient is

\({{\rm{\rho }}_{{\rm{X,Y}}}}{\rm{ = }}\frac{{{\rm{Cov(X,Y)}}}}{{{{\rm{\sigma }}_{\rm{X}}}{\rm{ \times }}{{\rm{\sigma }}_{\rm{Y}}}}}{\rm{ = }}\frac{{{\rm{ - 2/75}}}}{{\sqrt {{\rm{1/25}}} \sqrt {{\rm{1/25}}} }}\)

\( = - \frac{2}{3} \cdot \)

Therefore, the solution is \({\rho _{X,Y}} = - \frac{2}{3}\).