11.2 Problems on mathematical expectation  (Page 5/6)

(See Exercise 19 from "Problems On Random Vectors and Joint Distributions"). ${\displaystyle f_{XY}(t,u)=\frac{12}{179}(3t^2+u)}$ , for $0\le t\le 2$ , $0\le u\le min\{2,3-t\}$ .

(See Exercise 20 from "Problems On Random Vectors and Joint Distributions"). ${\displaystyle f_{XY}(t,u)=\frac{12}{227}(3t+2tu)}$ , for $0\le t\le 2$ , $0\le u\le min\{1+t,2\}$ .

(See Exercise 21 from "Problems On Random Vectors and Joint Distributions"). ${\displaystyle f_{XY}(t,u)=\frac{2}{13}(t+2u)}$ , for $0\le t\le 2$ , $0\le u\le min\{2t,3-t\}$ .

(See Exercise 22 from "Problems On Random Vectors and Joint Distributions"). ${\displaystyle f_{XY}(t,u)=I_{[0,1]}\left(t\right)\frac{3}{8}(t^2+2u)+I_{(1,2]}\left(t\right)\frac{9}{14}{t}^2{u}^2}$ ,

                                     for $0\le u\le 1$ .

The class $\{X,Y,Z\}$ of random variables is iid (independent, identically distributed) with common distribution

Let $W=3X-4Y+2Z$ . Determine $E\left[W\right]$ . Do this using icalc, then repeat with icalc3 and compare results.

(See Exercise 5 from "Problems on Functions of Random Variables") The cultural committee of a student organization has arranged a special deal for tickets to a concert. The agreement is that the organization will purchase tentickets at $20 each (regardless of the number of individual buyers). Additional tickets are available according to the following schedule:

11-20, $18 each; 21-30 $16 each; 31-50, $15 each; 51-100, $13 each

If the number of purchasers is a random variable X , the total cost (in dollars) is a random quantity $Z=g\left(X\right)$ described by

Suppose $X\sim$ Poisson (75). Approximate the Poisson distribution by truncating at 150. Determine $E\left[Z\right]$ and $E\left[Z^2\right]$ .