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Chapter 5: Q94SE (page 246)

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?

Short Answer

Expert verified

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

b. It is disturbing.

Step by step solution

01

Definition

The standard deviation is a measurement of a collection of values' variance or dispersion. A low standard deviation implies that the values are close to the set's mean, whereas a high standard deviation suggests that the values are dispersed over a larger range.

02

Calculating correlation coefficient

(a):

correlation coefficient

of \({\rm{X}}\) and \({\rm{Y}}\) is

\({{\rm{\rho }}_{{\rm{X,Y}}}}{\rm{ = }}\frac{{{\rm{Cov(X,Y)}}}}{{{{\rm{\sigma }}_{\rm{X}}}{\rm{ \times }}{{\rm{\sigma }}_{\rm{Y}}}}}\)

The variances of random variables \({\rm{X}}\) and \({\rm{Y}}\), because of the independence, are

\(\begin{aligned}{\rm{\sigma }}_{\rm{X}}^{\rm{2}}&= V(A + D)\\ &= V(A) + V(D)\\ &= \sigma _{\rm{A}}^{\rm{2}}{\rm{ + \sigma }}_{\rm{D}}^{\rm{2}}{\rm{,\sigma }}_{\rm{Y}}^{\rm{2}}\\&= V(B + E)\\&= V(B) + V(E)\\ &= \sigma _{\rm{B}}^{\rm{2}}{\rm{ + \sigma }}_{\rm{E}}^{\rm{2}}\end{aligned}\)

The covariance between random variables \({\rm{X}}\) and \({\rm{Y}}\) is

\(\begin{aligned}Cov(X,Y) &= Cov(A + D,B + E)\\ &= Cov(A,B) + Cov(A,E) + Cov(D,B) + Cov(D,E)\\ &= Cov(A,B) + 0 + 0 + 0\\ &= Cov(A,B){\rm{.}}\end{aligned}\)

The following stands for the correlation coefficient

\(\begin{aligned}{{\rm{\rho }}_{{\rm{X,Y}}}} &= \frac{{{\rm{Cov(X,Y)}}}}{{{{\rm{\sigma }}_{\rm{X}}}{\rm{ \times }}{{\rm{\sigma }}_{\rm{Y}}}}}\\ &= \frac{{{\rm{Cov(A,B)}}}}{{\sqrt {{\rm{\sigma }}_{\rm{A}}^{\rm{2}}{\rm{ + \sigma }}_{\rm{D}}^{\rm{2}}} \sqrt {{\rm{\sigma }}_{\rm{B}}^{\rm{2}}{\rm{ + \sigma }}_{\rm{E}}^{\rm{2}}} }}\\ &= \frac{{{\rm{Cov(A,B)}}}}{{\sqrt {{\rm{\sigma }}_{\rm{A}}^{\rm{2}}{\rm{ + \sigma }}_{\rm{D}}^{\rm{2}}} \sqrt {{\rm{\sigma }}_{\rm{B}}^{\rm{2}}{\rm{ + \sigma }}_{\rm{E}}^{\rm{2}}} }}{\rm{ \times }}\frac{{{{\rm{\sigma }}_{\rm{A}}}{{\rm{\sigma }}_{\rm{B}}}}}{{{{\rm{\sigma }}_{\rm{A}}}{{\rm{\sigma }}_{\rm{B}}}}}\\ &= \frac{{{\rm{Cov(A,B)}}}}{{{{\rm{\sigma }}_{\rm{A}}}{{\rm{\sigma }}_{\rm{B}}}}}{\rm{ \times }}\frac{{{{\rm{\sigma }}_{\rm{A}}}}}{{\sqrt {{\rm{\sigma }}_{\rm{A}}^{\rm{2}}{\rm{ + \sigma }}_{\rm{D}}^{\rm{2}}} }}{\rm{ \times }}\frac{{{{\rm{\sigma }}_{\rm{B}}}}}{{\sqrt {{\rm{\sigma }}_{\rm{B}}^{\rm{2}}{\rm{ + \sigma }}_{\rm{E}}^{\rm{2}}} }}\\ &= Corr(A,B) \times\sqrt {{\rm{Corr}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}} \right)} {\rm{ \times }}\sqrt {{\rm{Corr}}\left( {{{\rm{Y}}_{\rm{1}}}{\rm{,}}{{\rm{Y}}_{\rm{2}}}} \right)} \end{aligned}\)

03

Calculating correlation coefficient

Reminder for \({\rm{Corr}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}} \right)\) (replicate measurements):

The variances of random variables \({{\rm{X}}_{\rm{1}}}{\rm{ and }}{{\rm{X}}_{\rm{2}}}\)are

\(\begin{aligned}{\rm{V}}\left( {{{\rm{X}}_{\rm{1}}}} \right)&= V\left( {{\rm{W + }}{{\rm{E}}_{\rm{1}}}} \right)\\ &= V(W) + V\left( {{{\rm{E}}_{\rm{1}}}} \right)\\&= \sigma _{\rm{W}}^{\rm{2}}{\rm{ + \sigma }}_{\rm{E}}^{\rm{2}}\\{\rm{V}}\left( {{{\rm{X}}_{\rm{2}}}} \right) &= V \left( {{\rm{W + }}{{\rm{E}}_{\rm{2}}}} \right)\\ &= V(W) + V \left( {{{\rm{E}}_{\rm{1}}}} \right)\\ &= \sigma _{\rm{W}}^{\rm{2}}{\rm{ + \sigma }}_{\rm{E}}^{\rm{2}}\end{aligned}\)

and the covariance is

\(\begin{aligned}{\rm{Cov}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}} \right) &= Cov\left( {{\rm{W + }}{{\rm{E}}_{\rm{1}}}{\rm{,W + }}{{\rm{E}}_{\rm{2}}}} \right)\\ &= Cov(W,W) + Cov\left( {{\rm{W,}}{{\rm{E}}_{\rm{2}}}} \right){\rm{ + Cov}}\left( {{{\rm{E}}_{\rm{1}}}{\rm{,W}}} \right){\rm{ + Cov}}\left( {{{\rm{E}}_{\rm{1}}}{\rm{,}}{{\rm{E}}_{\rm{2}}}} \right)\\&= Cov(W,W) + 0 + 0 + 0\\&= \sigma _{\rm{W}}^{\rm{2}}\end{aligned}\)

Therefore, the correlation coefficient

\(\begin{array}{c}{{\rm{\rho }}_{{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}}}{\rm{ = }}\frac{{{\rm{Cov}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}} \right)}}{{{{\rm{\sigma }}_{{{\rm{X}}_{\rm{1}}}}}{\rm{ \times }}{{\rm{\sigma }}_{{{\rm{X}}_{\rm{2}}}}}}}\\{\rm{ = }}\frac{{{\rm{\sigma }}_{\rm{W}}^{\rm{2}}}}{{\sqrt {{\rm{\sigma }}_{\rm{W}}^{\rm{2}}{\rm{ + \sigma }}_{\rm{E}}^{\rm{2}}} \sqrt {{\rm{\sigma }}_{\rm{W}}^{\rm{2}}{\rm{ + \sigma }}_{\rm{E}}^{\rm{2}}} }}\\{\rm{ = }}\frac{{{\rm{\sigma }}_{\rm{W}}^{\rm{2}}}}{{{\rm{\sigma }}_{\rm{W}}^{\rm{2}}{\rm{ + \sigma }}_{\rm{E}}^{\rm{2}}}}{\rm{.}}\end{array}\)

The square root will be between \({\rm{0}}\) and \({\rm{1}}\), and the correlation will be lowered, because the correlation coefficient of duplicate measurements is always between \({\rm{0}}\) and\({\rm{1}}\).

04

Determining correlation is troubling

(b):

The product of the square root of correlations is

\(\begin{array}{l}\sqrt {{\rm{Corr}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}} \right)} {\rm{ \times }}\sqrt {{\rm{Corr}}\left( {{{\rm{Y}}_{\rm{1}}}{\rm{,}}{{\rm{Y}}_{\rm{2}}}} \right)} \\{\rm{ = }}\sqrt {{\rm{0}}{\rm{.81}}} {\rm{ \times }}\sqrt {{\rm{0}}{\rm{.9025}}} \\{\rm{ = 0}}{\rm{.855}}\end{array}\)

It is troubling that the measurement error reduces the correlation to such an amount.

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Most popular questions from this chapter

Young鈥檚 modulus is a quantitative measure of stiffness of an elastic material. Suppose that for aluminum alloy sheets of a particular type, its mean value and standard deviation are \({\rm{70 GPa}}\) and \({\rm{1}}{\rm{.6 GPa}}\), respectively (values given in the article 鈥淚nfluence of Material Properties Variability on Springback and Thinning in Sheet Stamping Processes: A Stochastic Analysis鈥� (Intl. J. of Advanced Manuf. Tech., \({\rm{2010:117 - 134}}\))).

  1. If \({\rm{\bar X}}\) is the sample mean young鈥檚 modulus for a random sample of \({\rm{n = 16}}\)sheets, where is the sampling distribution of \({\rm{\bar X}}\)centered, and what is the standard deviation of the \({\rm{\bar X}}\)distribution?
  2. Answer the questions posed in part (a) for a sample size of \({\rm{n = 64}}\)sheets.
  3. For which of the two random samples, the one of part (a) or the one of part (b), is \({\rm{\bar X}}\) more likely to be within \({\rm{1GPa}}\) of \({\rm{70 GPa}}\)? Explain your reasoning.

We have seen that if\(E\left( {{X_1}} \right) = E\left( {{X_2}} \right) = . . . = E\left( {{X_n}} \right) = \mu \), then\(E\left( {{X_1} + . . . + {X_n}} \right) = n\mu \). In some applications, the number of\({X_i} 's\)under consideration is not a fixed number n but instead is a rv N. For example, let N\(5\)be the number of components that are brought into a repair shop on a particular day, and let Xi denote the repair shop time for the\({i^{th}}\)component. Then the total repair time is\({X_1} + {X_2} + . . . + {X_n}\), the sum of a random number of random variables. When N is independent of the\({X_i} 's\), it can be shown that

\(E\left( {{X_1} + . . . + {X_N}} \right) = E\left( N \right) \times \mu \)

a. If the expected number of components brought in on a particular day is\({\bf{10}}\)and the expected repair time for a randomly submitted component is\({\bf{40}}\)min, what is the expected total repair time for components submitted on any particular day?

b. Suppose components of a certain type come in for repair according to a Poisson process with a rate of\(5\)per hour. The expected number of defects per component is\(3.5\). What is the expected value of the total number of defects on components submitted for repair during a four-hour period? Be sure to indicate how your answer follows from the general result just given.

If two loads are applied to a cantilever beam as shown in the accompanying drawing, the bending moment at \({\rm{0}}\) due to the loads is \({{\rm{a}}_{\rm{1}}}{{\rm{X}}_{\rm{1}}}{\rm{ + }}{{\rm{a}}_{\rm{2}}}{{\rm{X}}_{\rm{2}}}{\rm{ }}\)

a. Suppose that \({{\rm{X}}_{\rm{1}}}{\rm{ and }}{{\rm{X}}_{\rm{2}}}{\rm{ }}\)are independent rv鈥檚 with means \({\rm{2 and 4kip}}\), respectively, and standard deviations \({\rm{.5}}\) and \({\rm{1}}{\rm{.0kip}}\), respectively. If \({{\rm{a}}_{\rm{1}}}{\rm{ = 5ft and }}{{\rm{a}}_{\rm{2}}}{\rm{ = 10ft }}\), what is the expected bending moment and what is the standard deviation of the bending moment?

b. If \({{\rm{X}}_{\rm{1}}}{\rm{ and }}{{\rm{X}}_{\rm{2}}}{\rm{ }}\)are normally distributed, what is the probability that the bending moment will exceed 75 kip-ft?

c. Suppose the positions of the two loads are random variables. Denoting them by \({{\rm{A}}_{\rm{1}}}{\rm{ and }}{{\rm{A}}_{\rm{2}}}{\rm{ }}\), assume that these variables have means of \({\rm{5 and 10ft }}\), respectively, that each has a standard deviation of \({\rm{.5}}\), and that all are independent of one another. What is the expected moment now?

d. For the situation of part (c), what is the variance of the bending moment?

e. If the situation is as described in part (a) except that \({\rm{Corr}}\left( {{{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}} \right){\rm{ = 0}}{\rm{.5}}\) (so that the two loads are not independent), what is the variance of the bending moment?

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.

Refer to Exercise.

a. Calculate the covariance between \({X_1} = \)the number of customers in the express checkout and\({X_2} = \)the number of customers in the superexpress checkout.

b. Calculate\(V\left( {{X_1} + {X_2}} \right)\). How does this compare to\(V\left( {{X_1}} \right) + V\left( {{X_2}} \right)\)?
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