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Let X1, X2,... be the annual rainfalls in Boston (measured in inches) in the years 2101

Chapter 5, Problem 61

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QUESTION:

Let X1, X2,... be the annual rainfalls in Boston (measured in inches) in the years 2101, 2102, . . . , respectively. Assume that annual rainfalls are i.i.d. draws from a continuous distribution. A rainfall value is a record high if it is greater than those in all previous years (starting with 2101), and a record low if it is lower than those in all previous years. (a) In the 22nd century (the years 2101 through 2200, inclusive), find the expected number of years that have either a record low or a record high rainfall. (b) On average, in how many years in the 22nd century is there a record low followed in the next year by a record high? (c) By definition, the year 2101 is a record high (and record low). Let N be the number of years required to get a new record high. Find P(N>n) for all positive integers n, and use this to find the PMF of N.

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QUESTION:

Let X1, X2,... be the annual rainfalls in Boston (measured in inches) in the years 2101, 2102, . . . , respectively. Assume that annual rainfalls are i.i.d. draws from a continuous distribution. A rainfall value is a record high if it is greater than those in all previous years (starting with 2101), and a record low if it is lower than those in all previous years. (a) In the 22nd century (the years 2101 through 2200, inclusive), find the expected number of years that have either a record low or a record high rainfall. (b) On average, in how many years in the 22nd century is there a record low followed in the next year by a record high? (c) By definition, the year 2101 is a record high (and record low). Let N be the number of years required to get a new record high. Find P(N>n) for all positive integers n, and use this to find the PMF of N.

ANSWER:

Step 1 of 3

(a)

Define random variables  that indicates whether  th year has been record high or record low or not Observe that

since it is the first considered year, so it has to be record, and that

for  since the permutation that we are interested in are the permutations where  th year is at the first or the last place. Hence, we have that

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