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Chapter 7: Q.7.8 (page 359)

Show that Xis stochastically larger than Yif and only ifE[f(X)]鈮�E[f(Y)]

for all increasing functions f..

Hint: Show that X鈮�stY, then E[f(X)]鈮�E[f(Y)]by showing that f(X)鈮�stf(Y)and then using Theoretical Exercise 7.7. To show that if E[f(X)]鈮�E[f(Y)]for all increasing functions f, then P{X>t}鈮�P{Y>t}, define an appropriate increasing function f.

Short Answer

Expert verified

It has been show that Xis stochastically larger than Yif and only ifE[f(X)]鈮�E[f(Y)]for all increasing functionsf.

Step by step solution

01

Given Information

Xis stochastically larger than Yif and only ifE[f(X)]鈮�E[f(Y)].

02

Explanation

Case 1: IfX鈮�stY

Thenf(x)鈮�stf(y)(鈭�fis an increasing function)

E[f(X)]鈮�.E[f(X)](using the result of positive exercise)

03

Explanation

Case 2: If E[f(X)]鈮�E[f(X)]

E[f(X)]=鈭�-鈭�鈭�P[f(x)>t]dt

E[f(Y)]=鈭�-鈭�鈭�P[f(Y)>t]dt

As E[f(X)]鈮�E[f(Y)]

鈬�E[f(X)>t]鈮�P[f(Y)>t]鈭赌t

Asfis an increasing function

P[X>t]鈮�P[Y>t]

鈬�X鈮�stY

04

Final Answer

Hence, it has been shown that Xis stochastically larger than Yif and only ifE[f(X)]鈮�E[f(Y)].

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Most popular questions from this chapter

Suppose in Self-Test Problem 7.3that the 20people are to be seated at seven tables, three of which have 4seats and four of which have 2seats. If the people are randomly seated, find the expected value of the number of married couples that are seated at the same table.

Consider n independent flips of a coin having probability p of landing on heads. Say that a changeover occurs whenever an outcome differs from the one preceding it. For instance, if n=5 and the outcome isHHTHT, then there are 3 changeovers. Find the expected number of changeovers. Hint: Express the number of changeovers as the sum of n鈭�1 Bernoulli random variables.

Show that Zis a standard normal random variable and if Yis defined by Y=a+bZ+cZ2, then

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A bottle initially contains m large pills and n small pills. Each day, a patient randomly chooses one of the pills. If a small pill is chosen, then that pill is eaten. If a large pill is chosen, then the pill is broken in two; one part is returned to the bottle (and is now considered a small pill) and the other part is then eaten.

(a) Let X denote the number of small pills in the bottle after the last large pill has been chosen and its smaller half returned. Find E[X].

Hint: De铿乶e n + m indicator variables, one for each of the small pills initially present and one for each of the small pills created when a large one is split in two. Now use the argument of Example 2m.

(b) Let Y denote the day on which the last large pills chosen. Find E[Y].

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