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Chapter 5: Q70E (page 242)

Consider a random sample of size n from a continuous distribution having median \({\rm{0}}\)so that the probability of any one observation being positive is \({\rm{.5}}\). Disregarding the signs of the observations, rank them from smallest to largest in absolute value, and let \({\rm{w = }}\)the sum of the ranks of the observations having positive signs. For example, if the observations are \({\rm{ - }}{\rm{.3, + }}{\rm{.7, + 2}}{\rm{.1 and - 2}}{\rm{.5 }}\) then the ranks of positive observations are \({\rm{ 2and 3 }}\), so \({\rm{w = 5}}\). In Chapter \({\rm{15}}\), W will be called Wilcoxon鈥檚 signed-rank statistic. \({\rm{w}}\)can be represented as follows:

where the Yi 鈥檚 are independent Bernoulli rv鈥檚, each with \({\rm{ p = }}{\rm{.5 }}\) (\({{\rm{Y}}_{\rm{1}}}{\rm{ = 1}}\)corresponds to the observation with rank \({\rm{i }}\)being positive)

a. Determine \({\rm{E}}\left( {{{\rm{Y}}_{\rm{i}}}} \right){\rm{ }}\)and then \({\rm{E(W)}}\)using the equation for \({\rm{W}}\). (Hint: The first n positive integers sum to \({\rm{n(n + 1)/2}}{\rm{.)}}\)

b. Determine \({\rm{V(}}{{\rm{Y}}_{\rm{1}}}{\rm{)}}\)and then \({\rm{V(W)}}\). (Hint: The sum of the squares of the first n positive integers can be expressed as \({\rm{n(n + 1)(2n + 1)/6}}{\rm{.)}}\)

Short Answer

Expert verified

\(\begin{array}{l}{\rm{a}}{\rm{.\;E}}\left( {{{\rm{Y}}_{\rm{i}}}} \right){\rm{ = 0}}{\rm{.5;E(W) = }}\frac{{{\rm{n(n + 1)}}}}{{\rm{4}}}{\rm{;\;}}\\{\rm{b}}{\rm{.\;V}}\left( {{{\rm{Y}}_{\rm{i}}}} \right){\rm{ = 0}}{\rm{.25;V(W) = }}\frac{{{\rm{n(n + 1)(2n + 1)}}}}{{{\rm{24}}}}\end{array}\)

Step by step solution

01

Definition of probability

the proportion of the total number of conceivable outcomes to the number of options in an exhaustive collection of equally likely outcomes that cause a given occurrence.

02

Determining that \({\rm{E}}\left( {{{\rm{Y}}_{\rm{i}}}} \right)\)and then \({\rm{E(W)}}\)using the equation for \({\rm{W}}\)

The Bernoulli random variable has an expected value of \({\rm{0}}{\rm{.5}}\). This is because such a random variable can take either the value \({\rm{0}}\) or the value \({\rm{1}}\) with a probability of \({\rm{0}}{\rm{.5}}\), implying that

\({\rm{E}}\left( {{{\rm{Y}}_{\rm{i}}}} \right){\rm{ = 0 \times 0}}{\rm{.5 + 1 \times 0}}{\rm{.5 = 0}}{\rm{.5}}\)

$W$ is predicted to have a value of

\(\begin{aligned}{{}}{{\rm{E(W) = E}}\left( {{\rm{1 \times }}{{\rm{Y}}_{\rm{1}}}{\rm{ + 2 \times }}{{\rm{Y}}_{\rm{2}}}{\rm{ + \ldots + n \times }}{{\rm{Y}}_{\rm{n}}}} \right)}\\{{\rm{ = 1 \times E}}\left( {{{\rm{Y}}_{\rm{1}}}} \right){\rm{ + 2 \times E}}\left( {{{\rm{Y}}_{\rm{2}}}} \right){\rm{ + \ldots + n \times E}}\left( {{{\rm{Y}}_{\rm{n}}}} \right)}\\{{\rm{ = }}\mathop {\rm{{a}}}\limits_{{\rm{k = 1}}}^{\rm{n}} {\rm{nk \times E}}\left( {{{\rm{Y}}_{\rm{k}}}} \right){\rm{ = }}\mathop {\rm{{a}}}\limits_{{\rm{k = 1}}}^{\rm{n}} {\rm{nk \times 0}}{\rm{.5}}}\\{{\rm{ = 0}}{\rm{.5}}\mathop {\rm{{a}}}\limits_{{\rm{k = 1}}}^{\rm{n}} {\rm{nk = }}\frac{{{\rm{n(n + 1)}}}}{{\rm{4}}}{\rm{.}}}\end{aligned}\)

03

Determining \({\rm{V(Yi)}}\) and then\({\rm{  V(W)}}\)

Bernoulli's random variable has a variance of

\(\begin{array}{*{20}{c}}{{\rm{V}}\left( {{{\rm{Y}}_{\rm{i}}}} \right){\rm{ = E}}\left( {{\rm{Y}}_{\rm{i}}^{\rm{2}}} \right){\rm{ - }}{{\left( {{\rm{E}}\left( {{{\rm{Y}}_{\rm{i}}}} \right)} \right)}^{\rm{2}}}}\\{{\rm{ = }}{{\rm{0}}^{\rm{2}}}{\rm{ \times 0}}{\rm{.5 + }}{{\rm{1}}^{\rm{2}}}{\rm{ \times 0}}{\rm{.5 - 0}}{\rm{.}}{{\rm{5}}^{\rm{2}}}}\\{{\rm{ = 0}}{\rm{.25}}}\end{array}\)

Given that the random variables \({{\rm{Y}}_{\rm{i}}}\) are independent, the variance is as follows:

\(\begin{array}{*{20}{c}}{{\rm{V(W) = V}}\left( {{\rm{1 \times }}{{\rm{Y}}_{\rm{1}}}{\rm{ + 2 \times }}{{\rm{Y}}_{\rm{2}}}{\rm{ + \ldots + n \times }}{{\rm{Y}}_{\rm{n}}}} \right)}\\{{\rm{ = }}{{\rm{1}}^{\rm{2}}}{\rm{ \times V}}\left( {{{\rm{Y}}_{\rm{1}}}} \right){\rm{ + }}{{\rm{2}}^{\rm{2}}}{\rm{ \times V}}\left( {{{\rm{Y}}_{\rm{2}}}} \right){\rm{ + \ldots + }}{{\rm{n}}^{\rm{2}}}{\rm{ \times V}}\left( {{{\rm{Y}}_{\rm{n}}}} \right)}\\{{\rm{ = }}\mathop {\rm{{a}}}\limits_{{\rm{k = 1}}}^{\rm{n}} {\rm{n}}{{\rm{k}}^{\rm{2}}}{\rm{ \times V}}\left( {{{\rm{Y}}_{\rm{k}}}} \right){\rm{ = }}\mathop {\rm{{a}}}\limits_{{\rm{k = 1}}}^{\rm{n}} {\rm{nk \times 0}}{\rm{.25}}}\\{{\rm{ = 0}}{\rm{.25}}\mathop {\rm{{a}}}\limits_{{\rm{k = 1}}}^{\rm{n}} {\rm{n}}{{\rm{k}}^{\rm{2}}}{\rm{ = }}\frac{{{\rm{n(n + 1)(2n + 1)}}}}{{{\rm{24}}}}}\end{array}\)

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Most popular questions from this chapter

Suppose the proportion of rural voters in a certain state who favor a particular gubernatorial candidate is\(.{\bf{45}}\)and the proportion of suburban and urban voters favouring the candidate is\(.{\bf{60}}\). If a sample of\({\bf{200}}\)rural voters and\({\bf{300}}\)urban and suburban voters is obtained, what is the approximate probability that at least\(\;{\bf{250}}\)of these voters favour this candidate?

Three different roads feed into a particular freeway entrance. Suppose that during a fixed time period, the number of cars coming from each road onto the freeway is a random variable, with expected value and standard deviation as given in the table.

a. What is the expected total number of cars entering the freeway at this point during the period? (Hint: Let \({\rm{Xi = }}\)the number from road\({\rm{i}}\).)

b. What is the variance of the total number of entering cars? Have you made any assumptions about the relationship between the numbers of cars on the different roads?

c. With \({\rm{Xi}}\) denoting the number of cars entering from road\({\rm{i}}\)during the period, suppose that \({\rm{cov}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}} \right){\rm{ = 80\; and\; cov}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{3}}}} \right){\rm{ = 90\; and\; cov}}\left( {{{\rm{X}}_{\rm{2}}}{\rm{,}}{{\rm{X}}_{\rm{3}}}} \right){\rm{ = 100}}\) (so that the three streams of traffic are not independent). Compute the expected total number of entering cars and the standard deviation of the total.

A health-food store stocks two different brands of a certain type of grain. Let \(X = \)the amount (lb) of brand A on hand and \(Y = \)the amount of brand B on hand. Suppose the joint pdf of X and Y is

\(f(x,y) = \left\{ {\begin{array}{*{20}{c}}{kxy\;\;\;\;\;\;\;x \ge 0,\;y \ge 0,\;20 \le x + y \le 30}\\{0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;otherwise}\end{array}} \right\}\)

a. Draw the region of positive density and determine the value of k.

b. Are X and Y independent? Answer by first deriving the marginal pdf of each variable.

c. Compute \(P\left( {X + Y \le 25} \right)\).

d. What is the expected total amount of this grain on hand?

e. Compute \(Cov\left( {X, Y} \right)\)and\(Corr\left( {X, Y} \right)\).

f. What is the variance of the total amount of grain on hand?

A more accurate approximation to \({\rm{E}}\left( {{\rm{h}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{, \ldots ,}}{{\rm{X}}_{\rm{n}}}} \right)} \right)\) in Exercise 95 is

\(h\left( {{\mu _1}, \ldots ,{\mu _n}} \right) + \frac{1}{2}\sigma _1^2\left( {\frac{{{\partial ^2}h}}{{\partial x_1^2}}} \right) + \cdots + \frac{1}{2}\sigma _n^2\left( {\frac{{{\partial ^2}h}}{{\partial x_n^2}}} \right)\)

Compute this for \({\rm{Y = h}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}{\rm{,}}{{\rm{X}}_{\rm{3}}}{\rm{,}}{{\rm{X}}_{\rm{4}}}} \right)\)given in Exercise 93 , and compare it to the leading term \({\rm{h}}\left( {{{\rm{\mu }}_{\rm{1}}}{\rm{, \ldots ,}}{{\rm{\mu }}_{\rm{n}}}} \right)\).

Show that if, then\({\rm{Corr(X,Y) = + 1}}\) or\({\rm{ - 1}}\). Under what conditions will\({\rm{\rho = + 1}}\)?
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