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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

A box contains ten sealed envelopes numbered\({\rm{1, \ldots ,10}}\). The first five contain no money, the next three each contains\({\rm{\$ 5}}\), and there is a \({\rm{\$ 10}}\) bill in each of the last two. A sample of size \({\rm{3}}\) is selected with replacement (so we have a random sample), and you get the largest amount in any of the envelopes selected. If \({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}\), and \({{\rm{X}}_{\rm{3}}}\) denote the amounts in the selected envelopes, the statistic of interest is \({\rm{M = }}\) the maximum of\({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}\), and\({{\rm{X}}_{\rm{3}}}\).

a. Obtain the probability distribution of this statistic.

b. Describe how you would carry out a simulation experiment to compare the distributions of \({\rm{M}}\) for various sample sizes. How would you guess the distribution would change as \({\rm{n}}\) increases?

Suppose that when the pH of a certain chemical compound is\(5.00\), the pH measured by a randomly selected beginning chemistry student is a random variable with a mean of\(5.00\)and a standard deviation .2. A large batch of the compound is subdivided and a sample is given to each student in a morning lab and each student in an afternoon lab. Let\(X = \)the average pH as determined by the morning students and\(Y = \)the average pH as determined by the afternoon students.

a. If pH is a normal variable and there are\(25\)students in each lab, compute\(P\left( { - .1 \le X - Y \le - .1} \right)\)

b. If there are\(36\)students in each lab, but pH determinations are not assumed normal, calculate (approximately)\(P\left( { - .1 \le X - Y \le - .1} \right)\).

Carry out a simulation experiment using a statistical computer package or other software to study the sampling distribution of \({\rm{\bar X}}\) when the population distribution is Weibull with \({\rm{\alpha = 2}}\) and\({\rm{\beta = 5}}\), as in Example\({\rm{5}}{\rm{.20}}\).[A1] Consider the four sample sizes, and\({\rm{30}}\), and in each case use \({\rm{1000}}\) replications. For which of these sample sizes does the \({\rm{\bar X}}\) sampling distribution appear to be approximately normal?

There are \({\rm{40}}\) students in an elementary statistics class. On the basis of years of experience, the instructor knows that the time needed to grade a randomly chosen first examination paper is a random variable with an expected value of \({\rm{6}}\)min and a standard deviation of \({\rm{6}}\)min.

a. If grading times are independent and the instructor begins grading at \({\rm{6:50}}\) p.m. and grades continuously, what is the (approximate) probability that he is through grading before the \({\rm{11:00}}\) p.m. TV news begins?

b. If the sports report begins at \({\rm{11:10,}}\) what is the probability that he misses part of the report if he waits until grading is done before turning on the TV?

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?
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