10.4 Exercise 5 — Applied Probability

(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 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

g \left(X\right) = 200 + 18 I_{M 1} \left(X\right) \left(X - 10\right) + \left(16 - 18\right) I_{M 2} \left(X\right) \left(X - 20\right) +

\left(15 - 16\right) I_{M 3} \left(X\right) \left(X - 30\right) + \left(13 - 15\right) I_{M 4} \left(X\right) \left(X - 50\right)

\text{where} M 1 = \left[10 , \infty\right) , M 2 = \left[20 , \infty\right) , M 3 = \left[30 , \infty\right) , M 4 = \left[50 , \infty\right)

Suppose $X \sim$ Poisson (75). Approximate the Poisson distribution by truncating at 150. Determine $P \left(Z \geq 1000\right) , P \left(Z \geq 1300\right)$ , and $P \left(900 \leq Z \leq 1400\right)$ .

Pfeiffer, Applied Probability, 10.4 Problems on Functions of Random Variables, exercise 5

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