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Chapter 5: Q76SE (page 243)

In cost estimation, the total cost of a project is the sum of component task costs. Each of these costs is a random variable with a probability distribution. It is customary to obtain information about the total cost distribution by adding together characteristics of the individual component cost distributions鈥攖his is called the 鈥渞oll-up鈥� procedure. For example, \(E\left( {{X_1} + .. + {X_n}} \right) = E\left( {{X_1}} \right) + .. + E\left( {{X_n}} \right)\), so the roll-up procedure is valid for mean cost. Suppose that there are two component tasks that \({X_1} and {X_2}\)are independent, normally distributed random variables. Is the roll-up procedure valid for the \(7{5^{th}}\) percentile? That is, is the \(7{5^{th}}\) percentile of the distribution of\({X_1} + {X_2}\)the same as the sum of the \(7{5^{th}}\) percentiles of the two individual distributions? If not, what is the relationship between the percentile of the sum and the sum of percentiles? For what percentiles is the roll-up procedure valid in this case?

Short Answer

Expert verified

When \({\sigma _1} = 0\)or \({\sigma _2} = 0\)or both \({\sigma _1} = {\sigma _2} = 0\) or \({z_\alpha } = 0\). It does not stand for the \({75^{th}}\)percentile for the median.

Step by step solution

01

Definition of Probability Distribution

A probability distribution is a mathematical function in probability theory and statistics that offers the odds of distinct possible outcomes for an experiment.It's a mathematical representation of random phenomena in terms of sample space and event probability.

02

Calculation for the determination of percentile in the given question.

The roll-up procedure is valid when the sum of percentiles is equal to the percentile of sums.

Given random variables are normally distributed! Let the \(\alpha \)percentile of random variable \({X_1}\)be\({\mu _1} + {z_\alpha }{\sigma _1}\) where \({z_\alpha }\;is\;\alpha \)percentile of standard normal distribution, and let the \(\alpha \)percentile of random variable \({X_2}\) be \({\mu _2} + {z_\alpha }{\sigma _2}\)

Because of independence, the \(\alpha \)percentile of a random variable \({X_1} + {X_2}\)is

\(\begin{array}{c}E\left( {{X_1} + {X_2}} \right) + {z_\alpha }\sqrt {V\left( {{X_1} + {X_2}} \right)} = E\left( {{X_1}} \right) + E\left( {{X_2}} \right) + {z_\alpha }\sqrt {V\left( {{X_1}} \right) + V\left( {{X_2}} \right)} \\ = {\mu _1} + {\mu _2} + {z_\alpha }\sqrt {\sigma _1^2 + \sigma _2^2} \end{array}\)

03

Calculation for the determination of percentile in the given question.

Therefore, the sum of percentiles is \({\mu _1} + {z_\alpha }{\sigma _1} + {\mu _2} + {z_\alpha }{\sigma _2}\) and the percentile of the sum is

\({\mu _1} + {\mu _2} + {z_\alpha }\sqrt {\sigma _1^2 + \sigma _2^2} \)

The equality

\({\mu _1} + {z_\alpha }{\sigma _1} + {\mu _2} + {z_\alpha }{\sigma _2} = {\mu _1} + {\mu _2} + {z_\alpha }\sqrt {\sigma _1^2 + \sigma _2^2} \)

stands if \({\sigma _1} = 0\)or both \({\sigma _1} = {\sigma _2} = 0\) or \({z_\alpha } = 0\)

The roll-up procedure does not stand for the \({75^{{\rm{th }}}}\)percentile.

The only percentile for which the roll-up procedure is valid is the median, or \({z_{50}} = 0\).

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Most popular questions from this chapter

A restaurant serves three fixed-price dinners costing \({\rm{12, 15and 20}}\). For a randomly selected couple dining at this restaurant, let \({\rm{X = }}\)the cost of the man鈥檚 dinner and \({\rm{Y = }}\)the cost of the woman鈥檚 dinner. The joint \({\rm{pmf's}}\) of \({\rm{X and Y }}\)is given in the following table:

a. Compute the marginal \({\rm{pmf's}}\)of \({\rm{X and Y }}\)

b. What is the probability that the man鈥檚 and the woman鈥檚 dinner cost at most \({\rm{15}}\)each?

c. Are \({\rm{X and Y }}\)independent? Justify your answer.

d. What is the expected total cost of the dinner for the two people?

e. Suppose that when a couple opens fortune cookies at the conclusion of the meal, they find the message 鈥淵ou will receive as a refund the difference between the cost of the more expensive and the less expensive meal that you have chosen.鈥� How much would the restaurant expect to refund?

The time taken by a randomly selected applicant for a mortgage to fill out a certain form has a normal distribution with mean value \({\rm{10}}\) min and standard deviation \({\rm{2}}\) min. If five individuals fill out a form on one day and six on another, what is the probability that the sample average amount of time taken on each day is at most \({\rm{11}}\) min?

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.

Let \({\rm{A}}\)denote the percentage of one constituent in a randomly selected rock specimen, and let \({\rm{B}}\)denote the percentage of a second constituent in that same specimen. Suppose \({\rm{D}}\)and \({\rm{E}}\)are measurement errors in determining the values of \({\rm{A}}\)and \({\rm{B}}\)so that measured values are \({\rm{X = A + D}}\)and\({\rm{Y = B + E}}\), respectively. Assume that measurement errors are independent of one another and of actual values.

a. Show that

\({\rm{Corr(X,Y) = Corr(A,B) \times }}\sqrt {{\rm{Corr}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}} \right)} {\rm{ \times }}\sqrt {{\rm{Corr}}\left( {{{\rm{Y}}_{\rm{1}}}{\rm{,}}{{\rm{Y}}_{\rm{2}}}} \right)} \)

where \({{\rm{X}}_{\rm{1}}}\)and \({{\rm{X}}_{\rm{2}}}\)are replicate measurements on the value of\({\rm{A}}\), and \({{\rm{Y}}_{\rm{1}}}\)and \({{\rm{Y}}_{\rm{2}}}\)are defined analogously with respect to\({\rm{B}}\). What effect does the presence of measurement error have on the correlation?

b. What is the maximum value of \({\rm{Corr(X,Y)}}\)when \({\rm{Corr}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}} \right){\rm{ = }}{\rm{.8100}}\)and \({\rm{Corr}}\left( {{{\rm{Y}}_{\rm{1}}}{\rm{,}}{{\rm{Y}}_{\rm{2}}}} \right){\rm{ = }}{\rm{.9025?}}\)Is this disturbing?

Suppose the expected tensile strength of type-A steel is \({\rm{105ksi}}\)and the standard deviation of tensile strength is \({\rm{8ksi}}\). For type-B steel, suppose the expected tensile strength and standard deviation of tensile strength are \({\rm{100ksi}}\)and \({\rm{6ksi}}\), respectively. Let \({\rm{\bar X = }}\)the sample average tensile strength of a random sample of \({\rm{40}}\) type-A specimens, and let \({\rm{\bar Y = }}\)the sample average tensile strength of a random sample of \({\rm{35}}\)type-B specimens.

a. What is the approximate distribution of \({\rm{\bar X ? of \bar Y?}}\)

b. What is the approximate distribution of \({\rm{\bar X - \bar Y}}\)? Justify your answer.

c. Calculate (approximately) \(P( - 1拢\bar X - \bar Y拢1)\)

d. Calculate. If you actually observed , would you doubt that \({{\rm{\mu }}_{\rm{1}}}{\rm{ - }}{{\rm{\mu }}_{\rm{2}}}{\rm{ = 5?}}\)
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