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Chapter 4: Q14SE (page 273)

Suppose that\({{\bf{X}}_{\bf{0}}}{\bf{,}}{{\bf{X}}_{\bf{1}}}{\bf{, \ldots }}{{\bf{X}}_{\bf{n}}}\)are independent randomvariables, each having the same variance\({{\bf{\sigma }}^{\bf{2}}}\). Let\({{\bf{Y}}_{\bf{j}}}{\bf{ = }}{{\bf{X}}_{\bf{j}}}{\bf{ - }}{{\bf{X}}_{{\bf{j - 1}}}}\;{\bf{for}}\;{\bf{j = 1,}} \cdots {\bf{,n}}\), and let\({{\bf{\bar Y}}_{\bf{n}}}{\bf{ = }}\frac{{\bf{1}}}{{\bf{n}}}\sum\nolimits_{{\bf{j = 1}}}^{\bf{n}} {{{\bf{Y}}_{\bf{j}}}} \)Determine the value of\({\bf{Var}}\left( {{{{\bf{\bar Y}}}_{\bf{n}}}} \right)\).

Short Answer

Expert verified

The value of \(Var\left( {{{\bar Y}_n}} \right) = \frac{{2{\sigma ^2}}}{{{n^2}}}\) .

Step by step solution

01

Given information

Here\({X_1},{X_2}, \ldots {X_n}\)are some random variables. Each of them has the same variance,\({\sigma ^2}\). There are also a set of variables that is.\({Y_j} = {X_j} - {X_{j - 1}}\)for 箩=1,2鈥�,苍.

02

Define \({{\bf{\bar Y}}_{\bf{n}}}\)

Consider,

\(\begin{aligned}{}{{\bar Y}_n} = \frac{1}{n}\sum\nolimits_{j = 1}^n {{Y_j}} \\ = \frac{1}{n}\left( {\left( {{X_1} - {X_0}} \right) + \left( {{X_2} - {X_1}} \right) + \cdots + \left( {{X_n} - {X_{n - 1}}} \right)} \right)\\ = \frac{1}{n}\left( {{X_n} - {X_0}} \right)\end{aligned}\)

03

Derive the variance

The variance of the\({\bar Y_n}\)is given by,

\(\begin{aligned}{}Var\left( {{{\bar Y}_n}} \right) = \frac{1}{{{n^2}}}Var\left( {{X_n} - {X_0}} \right)\\ = \frac{1}{{{n^2}}}\left( {Var\left( {{X_n}} \right) + Var\left( {{X_0}} \right)} \right)\\ = \frac{1}{{{n^2}}}\left( {{\sigma ^2} + {\sigma ^2}} \right)\\ = \frac{{2{\sigma ^2}}}{{{n^2}}}\end{aligned}\)

Thus, we assume that \({X_n},{X_0}\) are independent. So, the required variance is \(\frac{{2{\sigma ^2}}}{{{n^2}}}\)

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Most popular questions from this chapter

Suppose that the joint distribution of X and Y is the uniform distribution on the circle\({{\bf{x}}^{\bf{2}}}{\bf{ + }}{{\bf{y}}^{\bf{2}}}{\bf{ < 1}}\). Find\({\bf{E}}\left( {{\bf{X}}\left| {\bf{Y}} \right.} \right)\).

Suppose that X and Y have a continuous joint distribution for which the joint p.d.f. is as follows:

\({\bf{f}}\left( {{\bf{x,y}}} \right){\bf{ = }}\left\{ {\begin{align}{}{{\bf{x + y}}}&{{\bf{for}}\,\,{\bf{0}} \le {\bf{x}} \le {\bf{1}}\,\,{\bf{and}}\,\,{\bf{0}} \le {\bf{y}} \le {\bf{1,}}}\\{\bf{0}}&{{\bf{otherwise}}{\bf{.}}}\end{align}} \right.\).

Find\({\bf{E}}\left( {{\bf{Y}}\left| {\bf{X}} \right.} \right)\)and\({\bf{Var}}\left( {{\bf{Y}}\left| {\bf{X}} \right.} \right)\).

Consider three gambles, X, Y, and Z, for which the probability distributions of the gains are as follows:

\(\begin{align}{}{\bf{Pr}}\left( {{\bf{X = 5}}} \right){\bf{ = Pr}}\left( {{\bf{X = 25}}} \right){\bf{ = }}{\raise0.7ex\hbox{\({\bf{1}}\)} \!\mathord{\left/ {\vphantom {{\bf{1}} {\bf{2}}}}\right.\ } \!\lower0.7ex\hbox{\({\bf{2}}\)}}{\bf{,}}\\{\bf{Pr}}\left( {{\bf{Y = 10}}} \right){\bf{ = Pr}}\left( {{\bf{Y = 20}}} \right){\bf{ = }}{\raise0.7ex\hbox{\({\bf{1}}\)} \!\mathord{\left/ {\vphantom {{\bf{1}} {\bf{2}}}}\right.\ } \!\lower0.7ex\hbox{\({\bf{2}}\)}}{\bf{,}}\\{\bf{Pr}}\left( {{\bf{Z = 15}}} \right){\bf{ = 1}}{\bf{.}}\end{align}\)

Suppose that a person鈥檚 utility function has the form\({\bf{U}}\left( {\bf{x}} \right){\bf{ = }}{{\bf{x}}^{\bf{2}}}\)for\({\bf{x > 0}}\). Which of the three gambles would she prefer?

Suppose that a person鈥檚 utility function is\({\bf{U}}\left( {\bf{x}} \right){\bf{ = }}{{\bf{x}}^{\bf{2}}}\)for\({\bf{x}} \ge {\bf{0}}\). Show that the person will always prefer to take a gamble in which she will receive a random gain of X dollars rather than receive the amount E(X) with certainty, where\({\bf{{\rm P}}}\left( {{\bf{X}} \ge {\bf{0}}} \right){\bf{ = 1}}\)and\({\bf{E}}\left( {\bf{X}} \right){\bf{ < }}\infty \)

Construct an example of a distribution for which the mean is finite but the variance is infinite.
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