91Ó°ÊÓ

If the random variables \(X\) and \(Y\) have the joint pdf $$ f_{X, Y}(x, y)=\left\\{\begin{array}{l} 8 x y, 0 \leq y \leq x \leq 1 \\ 0, \text { otherwise } \end{array}\right. $$ show that \(\operatorname{Cov}(X, Y)=\frac{8}{450} .\) Calculate \(\rho(X, Y) .\)

Let the random variable \(X\) take on the values \(1,2, \ldots, n\), each with probability \(1 / n .\) Define \(Y\) to be \(X^{2} .\) Find \(\rho(X, Y)\) and \(\lim _{n \rightarrow \infty} \rho(X, Y)\).

Many people believe that a salary bonus is a reward for good performance. The corporate world may have a different understanding. A random sample of thirty chief executive officers of large capitalization public companies recorded the cash bonus paid, \(x\) (in \(\$ 100,000\) ), and the performance of the company, \(y\), as measured by percentage change in company revenues. The following sums resulted. $$ \begin{gathered} \sum_{i=1}^{30} x_{i}=1,300.69 \quad \sum_{i=1}^{30} y_{i}=323 \\ \sum_{i=1}^{30} x_{i}^{2}=86,754.6939 \quad \sum_{i=1}^{30} y_{i}^{2}=11,881 \\\ \sum_{i=1}^{30} x_{i} y_{i}=7,807.36 \end{gathered} $$ Find the sample coefficient of correlation. What does this study say about the relationship between bonuses and performance?