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Chapter 6: Q.401 (page 401)
Suppose you roll a pair of fair, six-sided dice until you get doubles. Let T = the number of rolls it takes. Show that T is a geometric random variable (check the BITS)
All of the prerequisites for a geometric random variable are met. is a geometric random variable as a result.
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Buying stock You purchase a hot stock for . The stock either gains or loses each day, each with probability . Its returns on consecutive days are independent of each other. You plan to sell the stock after two days.
(a) What are the possible values of the stock after two days, and what is the probability for each value? What is the probability that the stock is worth more after two days than the you paid for it?
(b) What is the mean value of the stock after two days?
(Comment: You see that these two criteria give different answers to the question "Should I invest?")
While he was a prisoner of war during World War II, John Kerrich tossed a coin times. He gotheads. If the coin is perfectly balanced, the probability of a head is .
(a) Find the mean and the standard deviation of the number of heads in tosses, assuming the coin is perfectly balanced.
(b) Explain why the Normal approximation is appropriate for calculating probabilities involving the number of heads in tosses.
(c) Is there reason to think that Kerrich鈥檚 coin was not balanced? To answer this question, use a Normal distribution to estimate the probability that tossing a balanced coin times would give a count of heads at least this far from (that is, at least heads or no more than heads
The mean of is
(a)
(b)
(c)
(d)
(e)
90. Normal approximation To use a Normal distribution to approximate binomial probabilities, why do we require that both and be at least ?
Knees Patients receiving artificial knees often experience pain after surgery. The pain is measured on a subjective scale with possible values of 1 (low) to 5 (high). Let X be the pain score for a randomly selected patient. The following table gives part of the probability distribution for X.
| Value | |||||
| Probability | ? |
(a) Find
(b) If two patients who received artificial knees are chosen at random, what鈥檚 the probability that both of them report pain scores of or ? Show your work.
(c) Compute the mean and standard deviation of . Show your work.
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