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A time series consists of observations x₁, x2,...,x, taken on a random variable at equally- spaced points of time. If, of three consecutive observations x,-1, x, and x+1, x, is the smallest or largest of the three, then there is said to be a turning-point in the series at time t. Let S be the total number of turning-points in the series of n observations. (a) For the case n = 4, suppose that x₁, x2, x3 and x4 take the values 1, 2, 3 and 4 in some order. Write down all the permutations of these four numbers, and for each permutation write down the value which S would take if x₁ were the first number, x2 the second, and so on. If all permutations are equally likely, what is the probability function of S? same (b) Now suppose that x1, x2,..., X are independent observations, all from the continuous probability distribution. Show that the expected value of S is (n - 2).

A time series consists of observations x₁, x2,...,x, taken on a random variable at equally- spaced points of time. If, of three consecutive observations x,-1, x, and x+1, x, is the smallest or largest of the three, then there is said to be a turning-point in the series at time t. Let S be the total number of turning-points in the series of n observations. (a) For the case n = 4, suppose that x₁, x2, x3 and x4 take the values 1, 2, 3 and 4 in some order. Write down all the permutations of these four numbers, and for each permutation write down the value which S would take if x₁ were the first number, x2 the second, and so on. If all permutations are equally likely, what is the probability function of S? same (b) Now suppose that x1, x2,..., X are independent observations, all from the continuous probability distribution. Show that the expected value of S is (n - 2).

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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A time series consists of observations x₁, x2,...,x, taken on a random variable at equally- spaced points of time. If, of three consecutive observations x,-1, x, and x+1, x, is the smallest or largest of the three, then there is said to be a turning-point in the series at time. Let S be the total number of turning-points in the series of n observations. (a) For the case n = 4, suppose that x₁, x2, x3 and x4 take the values 1, 2, 3 and 4 in some order. Write down all the permutations of these four numbers, and for each permutation write down the value which S would take if x₁ were the first number, x2 the second, and so on. If all permutations are equally likely, what is the probability function of S? (b) Now suppose that x1, x2,..., X are independent observations, all from the same continuous probability distribution. Show that the expected value of S is (n -2).
Transcribed Image Text:A time series consists of observations x₁, x2,...,x, taken on a random variable at equally- spaced points of time. If, of three consecutive observations x,-1, x, and x+1, x, is the smallest or largest of the three, then there is said to be a turning-point in the series at time. Let S be the total number of turning-points in the series of n observations. (a) For the case n = 4, suppose that x₁, x2, x3 and x4 take the values 1, 2, 3 and 4 in some order. Write down all the permutations of these four numbers, and for each permutation write down the value which S would take if x₁ were the first number, x2 the second, and so on. If all permutations are equally likely, what is the probability function of S? (b) Now suppose that x1, x2,..., X are independent observations, all from the same continuous probability distribution. Show that the expected value of S is (n -2).
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