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Chapter 5: Q58E (page 241)

A shipping company handles containers in three different sizes:\(\left( 1 \right)\;27f{t^3}\;\left( {3 \times 3 \times 3} \right)\)\(\left( 2 \right) 125 f{t^3}, and \left( 3 \right)\;512 f{t^3}\). Let \({X_i}\left( {i = \;1, 2, 3} \right)\)denote the number of type i containers shipped during a given week. With \({\mu _i} = E\left( {{X_i}} \right)\)and\(\sigma _i^2 = V\left( {{X_i}} \right)\), suppose that the mean values and standard deviations are as follows:

\(\begin{array}{l}{\mu _1} = 200 {\mu _2} = 250 {\mu _3} = 100 \\{\sigma _1} = 10 {\sigma _2} = \,12 {\sigma _3} = 8\end{array}\)

a. Assuming that \({X_1}, {X_2}, {X_3}\)are independent, calculate the expected value and variance of the total volume shipped. (Hint:\(Volume = 27{X_1} + 125{X_2} + 512{X_3}\).)

b. Would your calculations necessarily be correct if \({X_i} 's\)were not independent?Explain.

Short Answer

Expert verified

The expected value is \(87,850\)and the variance of the volume is \(19,100,116\).

Step by step solution

01

Definition of Probability

The probability of an event happening is defined by probability. Many real-life circumstances need us to forecast the outcome of an occurrence. We may be certain or uncertain about the outcome of an event. In such instances, we say that the occurrence has a chance of occurring or not occurring.

02

Calculation for the determination of expected value and variance of volume.

(a):

The expected values of the volume

\({V_o} = 27{X_1} + 125{X_2} + 512{X_3}\)

is

\(\begin{aligned}E\left( {{V_o}} \right) &= E\left( {27{X_1} + 125{X_2} + 512{X_3}} \right)\\ &= 27E\left( {{X_1}} \right) + 125E\left( {{X_2}} \right) + 512E\left( {{X_3}} \right)\\ &= 27 \cdot 200 + 125 \cdot 250 + 512 \cdot 100\\ &= 87,850\end{aligned}\)

(1): this stands for any random variables,

(2): the expectations are given in the exercise.

The variance of the volume is

\(\begin{aligned}V\left( {{V_o}} \right) &= V\left( {27{X_1} + 125{X_2} + 512{X_3}} \right)\\ &= {27^2}V\left( {{X_1}} \right) + {125^2}V\left( {{X_2}} \right) + {512^2}V\left( {{X_3}} \right)\\ &= {27^2} \cdot {10^2} + {125^2} \cdot {12^2} + {512^2} \cdot {8^2}\\ &= 19,100,116\end{aligned}\)

03

Further Calculation for the determination of expected value and variance of volume.

(3): the given random variables are independent therefore this equality stands,

(4): the standard variations of the random variables are given in the exercise.

The variance of the volume is

\(\begin{aligned}V\left( {{V_o}} \right) &= V\left( {27{X_1} + 125{X_2} + 512{X_3}} \right)\\ &= {27^2}V\left( {{X_1}} \right) + {125^2}V\left( {{X_2}} \right) + {512^2}V\left( {{X_3}} \right)\\ &= {27^2} \cdot {10^2} + {125^2} \cdot {12^2} + {512^2} \cdot {8^2}\\ &= 19,100,116\end{aligned}\)

(3): the given random variables are independent therefore this equality stands,

(4): the standard variations of the random variables are given in the exercise.

04

Explanation is given for part b.

(b):

As mentioned in (a),the expected value would stay the same, no matter the independence. However,

the variance would change

the equality (3): does not stand when the random variables are not independent (covariances should be included).

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Most popular questions from this chapter

Question: The number of customers waiting for gift-wrap service at a department store is an rv X with possible values \({\rm{0,1,2,3,4}}\)and corresponding probabilities \({\rm{.1,}}{\rm{.2,}}{\rm{.3,}}{\rm{.25,}}{\rm{.15}}{\rm{.}}\)A randomly selected customer will have \({\rm{1,2}}\),or \({\rm{3}}\) packages for wrapping with probabilities \({\rm{.6,}}{\rm{.3,}}\)and \({\rm{.1,}}\)respectively. Let \({\rm{Y = }}\)the total number of packages to be wrapped for the customers waiting in line (assume that the number of packages submitted by one customer is independent of the number submitted by any other customer).

a. Determine \({\rm{P(X = 3,Y = 3)}}\), i.e., \({\rm{P(3,3)}}\).

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Show that when \({\rm{X}}\) and \({\rm{Y}}\) are independent, \({\rm{Cov(X,Y) = Corr(X,Y) = 0}}\).

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The National Health Statistics Reports dated Oct. \({\rm{22, 2008}}\), stated that for a sample size of \({\rm{277 18 - }}\)year-old American males, the sample mean waist circumference was \({\rm{86}}{\rm{.3cm}}\). A somewhat complicated method was used to estimate various population percentiles, resulting in the following values:

a. Is it plausible that the waist size distribution is at least approximately normal? Explain your reasoning. If your answer is no, conjecture the shape of the population distribution.

b. Suppose that the population mean waist size is \({\rm{85cm}}\)and that the population standard deviation is \({\rm{15cm}}\). How likely is it that a random sample of \({\rm{277}}\) individuals will result in a sample mean waist size of at least \({\rm{86}}{\rm{.3cm}}\)?

c. Referring back to (b), suppose now that the population mean waist size in \({\rm{82cm}}\).Now what is the (approximate) probability that the sample mean will be at least \({\rm{86}}{\rm{.3cm}}\)? In light of this calculation, do you think that \({\rm{82cm}}\)is a reasonable value for \({\rm{\mu }}\)?

An ecologist wishes to select a point inside a circular sampling region according to a uniform distribution (in practice this could be done by first selecting a direction and then a distance from the center in that direction). Let \({\rm{X}}\)=the \({\rm{x}}\) coordinate of the point selected and \({\rm{Y}}\)=the \({\rm{y}}\) coordinate of the point selected. If the circle is centered at \({\rm{(0,0)}}\)and has radius \({\rm{R}}\), then the joint pdf of \({\rm{X}}\)and \({\rm{Y}}\) is

\({\rm{f(x,y) = }}\left\{ {\begin{array}{*{20}{c}}{\frac{{\rm{1}}}{{{\rm{\pi }}{{\rm{R}}^{\rm{2}}}}}}&{{{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}{\rm{拢}}{{\rm{R}}^{\rm{2}}}}\\{\rm{0}}&{{\rm{\;otherwise\;}}}\end{array}} \right.\)

a. What is the probability that the selected point is within \(\frac{{\rm{R}}}{{\rm{2}}}\)of the center of the circular region? (Hint: Draw a picture of the region of positive density \({\rm{D}}\). Because \({\rm{f}}\)(\({\rm{x}}\), \({\rm{y}}\)) is constant on \({\rm{D}}\), computing a probability reduces to computing an area.)

b. What is the probability that both \({\rm{X and Y}}\)differ from 0 by at most\(\frac{{\rm{R}}}{{\rm{2}}}\)?

c. Answer part (b) for\(\frac{{\rm{R}}}{{\sqrt {\rm{2}} }}\)replacing\(\frac{{\rm{R}}}{{\rm{2}}}\)

d. What is the marginal pdf of \({\rm{X}}\)? Of \({\rm{Y}}\)? Are \({\rm{X and Y}}\)independent?
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