10.4 Problems on Functions of Random Variables

Suppose X is a nonnegative, absolutely continuous random variable. Let $Z = g \left(X\right) = C e^{- a X}$, where $a > 0 , C > 0$. Then $0 < Z \leq C$. Use properties of the exponential and natural log function to show that

Use the result of the exercise to show that if $X \sim$ exponential $\left(\lambda\right)$, then

Present value of future costs. Suppose money may be invested at an annual rate a, compounded continually. Then one dollar in hand now, has a value $e^{a x}$ at the end of x years. Hence, one dollar spent x years in the future has a present value$e^{- a x}$. Suppose a device put into operation has time to failure (in years) $X \sim$ exponential $\left(\lambda\right)$. If the cost of replacement at failure is C dollars, then the present value…

Optimal stocking of merchandise. A merchant is planning for the Christmas season. He intends to stock m units of a certain item at a cost of c per unit. Experience indicates demand can be represented by a random variable $D \sim$ Poisson $\left(\mu\right)$. If units remain in stock at the end of the season, they may be returned with recovery of r per unit. If demand exceeds the number originally ordered, extra units ma…

(See Example 2 from "Functions of a Random Variable") 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 each
• 31-50, $15 e…

For the pair $\left\{X , Y\right\}$ in the exercise, let $Z = g \left(X , Y\right) = 3 X^{2} + 2 X Y - Y^{2}$. Determine and plot the distribution function for Z.

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

Determine and plot the distribution function for W.

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

Determine $P \left(Z \leq 2\right)$

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

Determine $P \left(Z \leq 1 / 4\right)$.

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

Determine $P \left(Z \leq 1\right)$.

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

Determine $P \left(Z \leq 2\right)$.

The class $\left\{X , Y , Z\right\}$ is independent.

          $X = - 2 I_{A} + I_{B} + 3 I_{C}$. Minterm probabilities are (in the usual order)

          $Y = I_{D} + 3 I_{E} + I_{F} - 3$. The class $\left\{D , E , F\right\}$ is independent with

          Z has distribution

Determine $P \left(X^{2} + 3 X Y^{2} > 3 Z\right)$.

The simple random variable X has distribution

1. Plot the distribution function F_X and the quantile function Q_X.
2. Take a random sample of size $n = 10 , 000$. Compare the relative frequency for each value with the probability that value is taken on.