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.

(a):

Covariance

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

\({\rm{Cov(X,Y) = E((X - E(X))(Y - E(Y)))}}\)

or equally

\({\rm{Cov(X,Y) = }}\left\{ {\begin{aligned}{*{20}{l}}{\sum\limits_{\rm{x}} {\sum\limits_{\rm{y}}^{\rm{¥}} {{\rm{(x - E(}}} } {\rm{X))(y - E(Y)) \times p(x,y)}}}&{{\rm{,X and Y discrete, }}}\\{\int_{{\rm{ - ¥}}}^{\rm{¥}} {\int_{{\rm{ - ¥}}}^{\rm{¥}} {{\rm{(x - E(}}} } {\rm{X))(y - E(Y)) \times f(x,y)dxdy}}}&{{\rm{,X and Y continuous}}{\rm{. }}}\end{aligned}} \right.\)

where \({\rm{p(x,y)}}\) is pmf and \({\rm{f(x,y)}}\) pdf.

However, it requires more time to compute the covariance by the definition, this is why we will use the following proposition.

Proposition:

The following holds

\({\rm{Cov(X,Y) = E(XY) - E(X) \times E(Y)}}\)

From the exercise \({\rm{9}}\) we can see how does the marginal pdf's look like, therefore we can compute the expectations.

The following holds

\(\begin{aligned}{\rm{E(Y) = E(X)}}\\{\rm{ = }}\int_{{\rm{20}}}^{{\rm{30}}} {\rm{x}} {{\rm{f}}_{\rm{X}}}{\rm{(x)dx}}\\{\rm{ = }}\int_{{\rm{20}}}^{{\rm{30}}} {\rm{x}} \left( {{\rm{10K}}{{\rm{x}}^{\rm{2}}}{\rm{ + 0}}{\rm{.05}}} \right){\rm{dx}}\\{\rm{ = }}\left. {{\rm{10K}}\frac{{{{\rm{x}}^{\rm{3}}}}}{{\rm{3}}}} \right|_{{\rm{20}}}^{{\rm{30}}}{\rm{ + }}\left. {{\rm{0}}{\rm{.05}}\frac{{{{\rm{x}}^{\rm{2}}}}}{{\rm{2}}}} \right|_{{\rm{20}}}^{{\rm{30}}}\\{\rm{ = 25}}{\rm{.33}}{\rm{.}}\end{aligned}\)

We also need the following expectation

\(\begin{aligned}{\rm{E(XY) = }}\int_{{\rm{20}}}^{{\rm{30}}} {\int_{{\rm{20}}}^{{\rm{30}}} {\rm{x}} } {\rm{y \times K}}\left( {{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}} \right){\rm{dxdy}}\\{\rm{ = K}}\int_{{\rm{20}}}^{{\rm{30}}} {\int_{{\rm{20}}}^{{\rm{30}}} {{{\rm{x}}^{\rm{3}}}} } {\rm{ydydx + K}}\int_{{\rm{20}}}^{{\rm{30}}} {\int_{{\rm{20}}}^{{\rm{30}}} {{{\rm{y}}^{\rm{3}}}} } {\rm{xdxdy}}\\{\rm{ = K}}\int_{{\rm{20}}}^{{\rm{30}}} {{{\rm{x}}^{\rm{3}}}} \left( {\left. {\frac{{{{\rm{y}}^{\rm{2}}}}}{{\rm{2}}}} \right|_{{\rm{20}}}^{{\rm{30}}}} \right){\rm{dx + K}}\int_{{\rm{20}}}^{{\rm{30}}} {{{\rm{y}}^{\rm{3}}}} \left( {\left. {\frac{{{{\rm{x}}^{\rm{2}}}}}{{\rm{2}}}} \right|_{{\rm{20}}}^{{\rm{30}}}} \right){\rm{dy}}\\{\rm{ = }}\left. {\left( {\left. {\frac{{{{\rm{y}}^{\rm{2}}}}}{{\rm{2}}}} \right|_{{\rm{20}}}^{{\rm{30}}}} \right){\rm{K \times }}\frac{{{{\rm{x}}^{\rm{4}}}}}{{\rm{4}}}} \right|_{{\rm{20}}}^{{\rm{30}}}{\rm{ + }}\left. {\left( {\left. {\frac{{{{\rm{x}}^{\rm{2}}}}}{{\rm{2}}}} \right|_{{\rm{20}}}^{{\rm{30}}}} \right){\rm{K \times }}\frac{{{{\rm{y}}^{\rm{4}}}}}{{\rm{4}}}} \right|_{{\rm{20}}}^{{\rm{30}}}\\{\rm{ = 641}}{\rm{.447}}\end{aligned}\)

Now we can compute the covariance as follows

\(\begin{aligned}{\rm{Cov(X,Y) = E(XY) - E(X) \times E(Y)}}\\{\rm{ = 641}}{\rm{.447 - 25}}{\rm{.33 \times 25}}{\rm{.33}}\\{\rm{ = - 0}}{\rm{.1619}}{\rm{.}}\end{aligned}\)

Therefore, the covariance is \({\mathop{\rm Cov}\nolimits} (X,Y) = - 0.1619\).

(b):

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 to compute the variances in order to determine the correlation coefficient.

The Variance of \({\rm{X}}\), denoted by \({\rm{V(X)}}\left( {{\rm{\sigma }}_{\rm{X}}^{\rm{2}}} \right.\) or \(\left. {{{\rm{\sigma }}^{\rm{2}}}} \right)\) is

\(\begin{aligned}{\rm{V(X) = }}{{\rm{\sigma }}_{\rm{X}}}\\{\rm{ = E}}\left( {{{{\rm{(X - E(X))}}}^{\rm{2}}}} \right)\\{\rm{ = E}}\left( {{{\rm{X}}^{\rm{2}}}} \right){\rm{ - (E(X)}}{{\rm{)}}^{\rm{2}}}\end{aligned}\)

The Standard Deviation of \({\rm{X}}\) is

\({{\rm{\sigma }}_{\rm{X}}}{\rm{ = }}\sqrt {{\rm{\sigma }}_{\rm{X}}^{\rm{2}}} \)

The expectation of random variables \({\rm{X}}\) and \({\rm{Y}}\) has been calculated. The following is true

\(\begin{aligned}{c}{\rm{E}}\left( {{{\rm{Y}}^{\rm{2}}}} \right){\rm{ = E}}\left( {{{\rm{X}}^{\rm{2}}}} \right)\\{\rm{ = }}\int_{{\rm{20}}}^{{\rm{30}}} {\rm{x}} {{\rm{f}}_{\rm{X}}}{\rm{(x)dx}}\\{\rm{ = }}\int_{{\rm{20}}}^{{\rm{30}}} {{{\rm{x}}^{\rm{2}}}} \left( {{\rm{10K}}{{\rm{x}}^{\rm{2}}}{\rm{ + 0}}{\rm{.05}}} \right){\rm{dx}}\\{\rm{ = }}\left. {{\rm{10K}}\frac{{{{\rm{x}}^{\rm{4}}}}}{{\rm{4}}}} \right|_{{\rm{20}}}^{{\rm{30}}}{\rm{ + }}\left. {{\rm{0}}{\rm{.05}}\frac{{{{\rm{x}}^{\rm{3}}}}}{{\rm{3}}}} \right|_{{\rm{20}}}^{{\rm{30}}}\\{\rm{ = 649}}{\rm{.825}}{\rm{.}}\end{aligned}\)

The variance is

\(\begin{aligned}{\rm{V(Y) = V(X)}}\\{\rm{ = E}}\left( {{{\rm{X}}^{\rm{2}}}} \right){\rm{ - (E(X)}}{{\rm{)}}^{\rm{2}}}\\{\rm{ = 649}}{\rm{.825 - (25}}{\rm{.33}}{{\rm{)}}^{\rm{2}}}\\{\rm{ = 8}}{\rm{.2161}}\end{aligned}\)

Finally, the correlation coefficient is

\(\begin{aligned}{{\rm{\rho }}_{{\rm{X,Y}}}}{\rm{ = }}\frac{{{\rm{Cov(X,Y)}}}}{{{{\rm{\sigma }}_{\rm{X}}}{\rm{ \times }}{{\rm{\sigma }}_{\rm{Y}}}}}\\{\rm{ = }}\frac{{{\rm{ - 0}}{\rm{.1619}}}}{{\sqrt {{\rm{8}}{\rm{.2161}}} \sqrt {{\rm{8}}{\rm{.2161}}} }}\\{\rm{ = - 0}}{\rm{.0024}}{\rm{.}}\end{aligned}\)

Therefore, the correlation coefficient is \({\rho _{X,Y}} = - 0.0024\).