Chapter 5.8, Problem 109E

In Exercise 5.16, Y₁ and Y₂ denoted the proportions of time that employees I and II actually spent working on their assigned tasks during a workday. The joint density of Y₁ and Y₂ is given by

$f\text{(}{y}_1,y_2\text{)}\text{=}\{\begin{array}{l}{y}_1+y_2,0\le y_1\le 1,0\le y_2\le 1,\hfill \\ 0,\text{elsewhere}.\hfill \end{array}$

In Exercise 5.80, we derived the mean of the productivity measure 30Y₁ + 25Y₂. Find the variance of this measure of productivity. Give an interval in which you think the total productivity measures of the two employees should lie for at least 75% of the days in question.

5.16 Let Y₁ and Y₂ denote the proportions of time (out of one workday) during which employees I and II, respectively, perform their assigned tasks. The joint relative frequency behavior of Y₁ and Y₂ is modeled by the density function

$f\text{(}{y}_1,y_2\text{)}\text{=}\{\begin{array}{l}{y}_1+y_2,0\le y_1\le 1,0\le y_2\le 1,\hfill \\ 0,\text{elsewhere}.\hfill \end{array}$

a Find P(Y₁ < 1/2, Y₂ > 1/4).

b Find P(Y₁ + Y₂ ≤ 1).

5.80 In Exercise 5.16, Y₁ and Y₂ denoted the proportions of time that employees I and II actually spent working on their assigned tasks during a workday. The joint density of Y₁ and Y₂ is given by

$f\text{(}{y}_1,y_2\text{)}\text{=}\{\begin{array}{l}{y}_1+y_2,0\le y_1\le 1,0\le y_2\le 1,\hfill \\ 0,\text{elsewhere}.\hfill \end{array}$

Employee I has a higher productivity rating than employee II and a measure of the total productivity of the pair of employees is 30Y₁ + 25Y₂. Find the expected value of this measure of productivity.