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.

The

Covariance

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

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

or equally

\({\mathop{\rm Cov}\nolimits} (X,Y) = \left\{ {\begin{array}{*{20}{l}}{\sum\limits_x {\sum\limits_y {(x - E(} } X))(y - E(Y)) \cdot p(x,y)}\\{\int_{ - \infty }^\infty {\int_{ - \infty }^\infty {(x - E(} } X))(y - E(Y)) \cdot f(x,y)dxdy}\end{array}} \right.\)

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

Proposition:

The following holds

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

We can find the marginal pdf of \({\rm{X}}\) and\({\rm{Y}}\), therefore we can compute expectations. The following is true

\(\begin{aligned}E(Y) &= \int_0^\infty y \cdot \frac{1}{{{{(1 + y)}^2}}}dy\\ &= \int_0^\infty {\frac{{(y + 1) - 1}}{{{{(1 + y)}^2}}}} dy\\ = \int_0^\infty {\frac{1}{{1 + y}}} dy - \int_0^\infty {\frac{1}{{{{(1 + y)}^2}}}} dy\end{aligned}\)

We have that

\(\int_0^\infty {\frac{1}{{1 + y}}} dy = \left. {\ln |1 + y|} \right|_0^\infty = \infty ,\)

This indicates that the probability is not finite. Both the covariance and the correlation coefficient are thus unknown.