Math 143 C/E, Spring 2001
IPS Reading Questions
Chapter 4, Section 4



  1. In Example 4.19 (p. 326) the random variable X = winnings (in dollars) from a Tri-State Pick 3 lottery ticket. The expected value (or mean) of this random variable is $0.50. Just what is meant here by the words “expected value”? Does your interpretation hold up to Example 4.20, where the random variable X = # of heads obtained from four coin tosses has an expected value of 2? Does it hold up to Example 4.21, where X = size of a household has an expected value of 2.6?

    The expected value is not the outcome you should expect for a random variable from a single iteration of the random phenomenon it describes. That is, you should not expect a payoff of $0.50 on one Tri-State Pick 3 lottery ticket; in fact, the payoff (to a single ticket-holder) is always either nothing at all or $500. Rather, if you keep a running total of the random variable's value over the course of many iterations of the same phenomenon (many lottery tickets purchased, for instance) and divide by the number of iterations, you should get something close to the expected value. This means that the state should go into this venture expecting that the total money paid out to lottery winners divided by the number of players should be about $0.50 per player. That's not too bad if players are charged $1.00 per ticket.

    A similar interpretation of the expected value of a random variable can be applied to both Examples 4.20 and 4.21.

  2. We showed in class that the expected value (mean) of the random variable X = sum of pips on two dice is 7. With two dice of your own and enough time on your hands, how might you illustrate the law of large numbers?

    You could roll the dice many times, each time keeping track of the roll (the total of pips). After, say, 1000 rolls, you could calculate the average = (total of all rolls)/(number of rolls) and demonstrate that this average was close to 7. You could continue on to 10000 rolls and show that the average was now (probably) closer to 7. Fifty-thousand rolls would generally yield an average even closer to 7.

  3. What examples on pp. 331-332 do the IPS authors give to substantiate the claim that we are unable “to accurately distinguish random behavior from systematic influences” (this quote comes from p. 332, about half-way down the page)?

    In the last paragraph on p. 331, they say that most people tend to think a truly random set of 10 coin tosses will not contain any runs of heads (nor of tails) longer than two when, in fact, the probability that such a run will occur for a fair coin is greater than 0.8. In the next paragraph, they point out how people look at streaks in athletics (streaks of made free-throws, streaks of consecutive games where a batter gets a hit) as an indication that the next opportunity (the next free-throw attempt, plate-appearances in the next softball game, etc.) is somehow influenced by or dependent upon recent performance (the “hot hand” theory), when actually one would expect such streaks even when each opportunity is treated as independent of previous ones.

  4. The heights (in inches) of American men ages 18-74 are generally distributed as N(67, 3), while those of women in the same age group are N(64.5, 2.5). Let us associate the random variable X with the heights of men and Y with the heights of women, and let Z = X+Y. Would you think the distribution of Z appropriate to describe the combined heights of married couples? Why or why not?

    Since the random variables X and Y are independent (the height of a randomly-chosen male should not affect the height of a randomly-chosen female), Z shows the distribution of the combined heights of one randomly-chosen male and one randomly-chosen female. Nevertheless, people do not choose their spouses randomly, and hence this distribution may not be appropriate for the combined heights of spouses.



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On 21 Feb 2001, 08:04.