By Sheldon M. Ross
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Extra info for An Elementary Introduction to Mathematical Finance, Third Edition
T) has mean To see why this might be the case, note that X (t+h)−X h μ and variance σ 2 / h. Because the variance of this ratio is converging to infinity as h → 0, it is not surprising that the ratio does not converge. 2 Brownian Motion as a Limit of Simpler Models Let be a small increment of time, and consider a process such that every time units the value of the process either increases √ by the amount √ with probability p or decreases by the amount σ with probaσ bility 1 − p, where 1 μ√ p= 1+ 2 σ and where the successive changes in value are independent.
B) What is the probability that the second battery will outlive the first by at least 25 hours? (c) What is the probability that the longer-lasting battery will outlive the other by at least 25 hours? 7 The time it takes to develop a photographic print is a random variable with mean 18 seconds and standard deviation 1 second. Approximate the probability that the total amount of time that it takes to process 100 prints is (a) more than 1,710 seconds; (b) between 1,690 and 1,710 seconds. 8 Frequent fliers of a certain airline fly a random number of miles each year, having mean and standard deviation of 25,000 and 12,000 miles, respectively.
Suppose that each set is equally likely to be won by either player, and that the results from different sets are independent. Find (a) the expected value and (b) the variance of the number of sets played. 11 Verify that Var(X ) = E[X 2 ] − (E[X ])2 . Hint: Starting with the definition Var(X ) = E[(X − E[X ])2 ], square the expression on the right side; then use the fact that the expected value of a sum of random variables is equal to the sum of their expectations. 12 A lawyer must decide whether to charge a fixed fee of $5,000 or take a contingency fee of $25,000 if she wins the case (and 0 if she loses).
An Elementary Introduction to Mathematical Finance, Third Edition by Sheldon M. Ross