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11.3 Exercise 34

Avg difficulty 2.5

(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 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(X)Z = g \left(X\right) described by

g ( X ) = 200 + 18 I M 1 ( X ) ( X - 10 ) + ( 16 - 18 ) I M 2 ( X ) ( X - 20 ) + 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) +
( 15 - 16 ) I M 3 ( X ) ( X - 30 ) + ( 13 - 15 ) I M 4 ( X ) ( X - 50 ) \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)
where M 1 = [ 10 , ∞ ) , M 2 = [ 20 , ∞ ) , M 3 = [ 30 , ∞ ) , M 4 = [ 50 , ∞ ) \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∼X \sim Poisson (75). Approximate the Poisson distribution by truncating at 150. Determine E[Z]E \left[Z\right] and E[Z2]E \left[Z^{2}\right].

Pfeiffer, Applied Probability, 11.3 Problems on Mathematical Expectation, exercise 34

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Adapted from Applied Probability © 2009 Paul E. Pfeiffer. Licensed under CC BY 3.0. Source: Applied Probability. Imported at revision d7a0420.