11.3 Problems on Mathematical Expectation

(See Exercise 5 from "Problems on Distribution and Density Functions"). In a thunderstorm in a national park there are 127 lightning strikes. Experience shows that the probability of of a lightning strike starting a fire is about 0.0083. Determine the expected number of fires.

(See Exercise 8 from "Problems on Distribution and Density Functions"). Two coins are flipped twenty times. Let X be the number of matches (both heads or both tails). Determine $E \left[X\right]$.

(See Exercise 12 from "Problems on Distribution and Density Functions"). A residential College plans to raise money by selling “chances” on a board. Fifty chances are sold. A player pays $10 to play; he or she wins $30 with probability $p = 0 . 2$. The profit to the College is

Determine the expected profit $E \left[X\right]$.

(See Exercise 19 from "Problems on Distribution and Density Functions"). The number of noise pulses arriving on a power circuit in an hour is a random quantity having Poisson (7) distribution. What is the expected number of pulses in an hour?

(See Exercise 24 and Exercise 25 from "Problems on Distribution and Density Functions"). The total operating time for the units in Exercise 24 is a random variable $T \sim$ gamma (20, 0.0002). What is the expected operating time?

(See Exercise 41 from "Problems on Distribution and Density Functions"). Random variable X has density function

What is the expected value $E \left[X\right]$?

Truncated exponential. Suppose $X \sim$ exponential $\left(\lambda\right)$ and $Y = I_{\left[0 , a\right]} \left(X\right) X + I_{\left(a , \infty\right)} \left(X\right) a$.

1. Use the fact that
   $\int_{0}^{\infty} t e^{- \lambda t} d t = \frac{1}{\lambda^{2}} \text{and} \int_{a}^{\infty} t e^{- \lambda t} d t = \frac{1}{\lambda^{2}} e^{- \lambda a} \left(1 + \lambda a\right)$
   to determine an expression for $E \left[Y\right]$.
2. Use the approximation method, with $\lambda = 1 / 50 , a = 30$. Approximate the exponential at 10,000 points for $0 \leq t \leq 1000$. Compare the approximate result with the theoretical result of part (a).

(See Exercise 10 from "Problems On Random Vectors and Joint Distributions"). $f_{X Y} \left(t , u\right) = 1$ for $0 \leq t \leq 1$, $0 \leq u \leq 2 \left(1 - t\right)$.

(See Exercise 11 from "Problems On Random Vectors and Joint Distributions"). $f_{X Y} \left(t , u\right) = 1 / 2$ on the square with vertices at $\left(1 , 0\right) , \left(2 , 1\right) , \left(1 , 2\right) , \left(0 , 1\right)$.

(See Exercise 12 from "Problems On Random Vectors and Joint Distributions"). $f_{X Y} \left(t , u\right) = 4 t \left(1 - u\right)$ for $0 \leq t \leq 1$, $0 \leq u \leq 1$.

(See Exercise 13 from "Problems On Random Vectors and Joint Distributions"). $f_{X Y} \left(t , u\right) = \frac{1}{8} \left(t + u\right)$ for $0 \leq t \leq 2$, $0 \leq u \leq 2$.

(See Exercise 14 from "Problems On Random Vectors and Joint Distributions"). $f_{X Y} \left(t , u\right) = 4 u e^{- 2 t}$ for $0 \leq t , 0 \leq u \leq 1$.

(See Exercise 15 from "Problems On Random Vectors and Joint Distributions"). $f_{X Y} \left(t , u\right) = \frac{3}{88} \left(2 t + 3 u^{2}\right)$ for $0 \leq t \leq 2$, $0 \leq u \leq 1 + t$.

(See Exercise 16 from "Problems On Random Vectors and Joint Distributions"). $f_{X Y} \left(t , u\right) = 12 t^{2} u$ on the parallelogram with vertices

(See Exercise 17 from "Problems On Random Vectors and Joint Distributions"). $f_{X Y} \left(t , u\right) = \frac{24}{11} t u$ for $0 \leq t \leq 2$, $0 \leq u \leq \min \left\{1 , 2 - t\right\}$.

(See Exercise 18 from "Problems On Random Vectors and Joint Distributions"). $f_{X Y} \left(t , u\right) = \frac{3}{23} \left(t + 2 u\right)$ for $0 \leq t \leq 2$, $0 \leq u \leq \max \left\{2 - t , t\right\}$.

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

(See Exercise 20 from "Problems On Random Vectors and Joint Distributions"). $f_{X Y} \left(t , u\right) = \frac{12}{227} \left(3 t + 2 t u\right)$, for $0 \leq t \leq 2$, $0 \leq u \leq \min \left\{1 + t , 2\right\}$.

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

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

                                     for $0 \leq u \leq 1$.

(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 ten tickets 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 eac…

For the pair $\left\{X , Y\right\}$ in the exercise, let

Determine $E \left[W\right]$ and $E \left[W^{2}\right]$.

$f_{X Y} \left(t , u\right) = \frac{3}{88} \left(2 t + 3 u^{2}\right)$ for $0 \leq t \leq 2$, $0 \leq u \leq 1 + t$ (see the exercise).

$f_{X Y} \left(t , u\right) = \frac{24}{11} t u$ for $0 \leq t \leq 2$, $0 \leq u \leq \min \left\{1 , 2 - t\right\}$ (see the exercise).

$f_{X Y} \left(t , u\right) = \frac{3}{23} \left(t + 2 u\right)$ for $0 \leq t \leq 2$, $0 \leq u \leq \max \left\{2 - t , t\right\}$ (see the exercise).

$f_{X Y} \left(t , u\right) = \frac{12}{179} \left(3 t^{2} + u\right)$, for $0 \leq t \leq 2$, $0 \leq u \leq \min \left\{2 , 3 - t\right\}$ (see the exercise).

$f_{X Y} \left(t , u\right) = \frac{12}{227} \left(3 t + 2 t u\right)$, for $0 \leq t \leq 2$, $0 \leq u \leq \min \left\{1 + t , 2\right\}$ (see the exercise).