10.3 Problems on functions of random variables

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

Use the result of [link] 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 $e^{ax}$ at the end of x years. Hence, one dollar spent x years in the future has a present value $e^{-ax}$ . 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 of the replacement is $Z=C{e}^{-aX}$ . Suppose $\lambda =1/10$ , $a=0.07$ , and $C=\$1000$ .

1. Use the result of [link] to determine the probability $Z\le 700,500,200$ .
2. Use a discrete approximation for the exponential density to approximate the probabilities in part (a). Truncate X at 1000 and use 10,000 approximation points.

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 may be ordered at a cost of s each. Units are sold at a price p per unit. If $Z=g\left(D\right)$ is the gain from the sales, then

• For $t\le m,g\left(t\right)=(p-c)t-(c-r)(m-t)=(p-r)t+(r-c)m$
• For $t>m,g\left(t\right)=(p-c)m+(t-m)(p-s)=(p-s)t+(s-c)m$

Let $M=(-\infty ,m]$ . Then

Suppose $\mu =50m=50c=30p=50r=20s=40.$ .
Approximate the Poisson random variable D by truncating at 100. Determine $P(500\le Z\le 1100)$ .

(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 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 $P(Z\ge 1000),P(Z\ge 1300)$ , and $P(900\le Z\le 1400)$ .