SUPPLEMENTARY EXERCISES (98-128)

98

Let $X=$ the time it takes a read/write head to locate a desired record on a computer disk memory device once the head has been positioned over the correct track. If the disks rotate once every 25 millisec, a reasonable assumption is that $X$ is uniformly distributed on the interval $\left[0,25\right]$ .

a. Compute $P\left(10\le X\le 20\right)$ .

b. Compute $P\left(X\ge 10\right)$ .

c. Obtain the cdf $F\left(X\right)$ .

d. Compute $E\left(X\right)$ and ${\sigma }_X$ .

99

A 12-in. bar that is clamped at both ends is to be subjected to an increasing amount of stress until it snaps. Let $Y=$ the distance from the left end at which the break occurs. Suppose $Y$ has pdf

$$f\left(y\right)=\{\begin{array}{cc}\left(\frac{1}{24}\right)y\left(1-\frac{y}{12}\right)& 0\le y\le 12\\ 0& \text{ otherwise }\end{array}$$

Compute the following:

a. The cdf of $Y$ , and graph it.

b. $P\left(Y\le 4\right),P\left(Y>6\right)$ , and $P\left(4\le Y\le 6\right)$

c. $E\left(Y\right),E\left(Y^2\right)$ , and $V\left(Y\right)$

d. The probability that the break point occurs more than 2 in. from the expected break point.

e. The expected length of the shorter segment when the break occurs.

100

Let $X$ denote the time to failure (in years) of a certain hydraulic component. Suppose the pdf of $X$ is $f\left(x\right)=$ $32/{\left(x+4\right)}^3$ for $x<0$ .

a. Verify that $f\left(x\right)$ is a legitimate pdf.

b. Determine the cdf.

c. Use the result of part (b) to calculate the probability that time to failure is between 2 and 5 years.

d. What is the expected time to failure?

e. If the component has a salvage value equal to $100/\left(4+x\right)$ when its time to failure is $x$ , what is the expected salvage value?

101

The completion time $X$ for a certain task has cdf $F\left(x\right)$ given by

$$\{\begin{array}{cc}0& x<0\\ \frac{x^3}{3}& 0\le x<1\\ 1-\frac{1}{2}\left(\frac{7}{3}-x\right)\left(\frac{7}{4}-\frac{3}{4}x\right)& 1\le x\le \frac{7}{3}\\ 1& x>\frac{7}{3}\end{array}$$

a. Obtain the $\mathrm{pdf}f\left(x\right)$ and sketch its graph.

b. Compute $P\left(.5\le X\le 2\right)$ .

c. Compute $E\left(X\right)$ .

102

Let $X$ represent the number of individuals who respond to a particular online coupon offer. Suppose that $X$ has approximately a Weibull distribution with $\alpha =10$ and $\beta =20$ . Calculate the best possible approximation to the probability that $X$ is between 15 and 20, inclusive.

103

The article “Computer Assisted Net Weight Control” (Quality Progress, 1983: 22-25) suggests a normal distribution with mean 137.2 oz and standard deviation 1.6 oz for the actual contents of jars of a certain type. The stated contents was $135\mathrm{oz}$ .

a. What is the probability that a single jar contains more than the stated contents?

b. Among ten randomly selected jars, what is the probability that at least eight contain more than the stated contents?

c. Assuming that the mean remains at 137.2, to what value would the standard deviation have to be changed so that $95\%$ of all jars contain more than the stated contents?

104

When circuit boards used in the manufacture of compact disc players are tested, the long-run percentage of defectives is $5\%$ . Suppose that a batch of 250 boards has been received and that the condition of any particular board is independent of that of any other board.

a. What is the approximate probability that at least $10\%$ of the boards in the batch are defective?

b. What is the approximate probability that there are exactly 10 defectives in the batch?

105

Exercise 38 introduced two machines that produce wine corks, the first one having a normal diameter distribution with mean value $3\mathrm{cm}$ and standard deviation $.1\mathrm{cm}$ , and the second having a normal diameter distribution with mean value $3.04\mathrm{cm}$ and standard deviation $.02\mathrm{cm}$ . Acceptable corks have diameters between 2.9 and $3.1\mathrm{cm}$ . If $60\%$ of all corks used come from the first machine and a randomly selected cork is found to be acceptable, what is the probability that it was produced by the first machine?

106

The reaction time (in seconds) to a certain stimulus is a continuous random variable with pdf

$$f\left(x\right)=\{\begin{array}{cc}\frac{3}{2}\cdot \frac{1}{x^2}& 1\le x\le 3\\ 0& \text{ otherwise }\end{array}$$

a. Obtain the cdf.

b. What is the probability that reaction time is at most 2.5 sec? Between 1.5 and 2.5 sec?

c. Compute the expected reaction time.

d. Compute the standard deviation of reaction time.

e. If an individual takes more than 1.5 sec to react, a light comes on and stays on either until one further second has elapsed or until the person reacts (whichever happens first). Determine the expected amount of time that the light remains lit.

Hint: Let $h\left(X\right)=$ the time that the light is on as a function of reaction time $X$ .

107

Let $X$ denote the temperature at which a certain chemical reaction takes place. Suppose that $X$ has pdf

$$f\left(x\right)=\{\begin{array}{cc}\frac{1}{9}\left(4-x^2\right)& -1\le x\le 2\\ 0& \text{ otherwise }\end{array}$$

a. Sketch the graph of $f\left(x\right)$ .

b. Determine the cdf and sketch it.

c. Is 0 the median temperature at which the reaction takes place? If not, is the median temperature smaller or larger than 0 ?

d. Suppose this reaction is independently carried out once in each of ten different labs and that the pdf of reaction time in each lab is as given. Let $Y=$ the number among the ten labs at which the temperature exceeds 1. What kind of distribution does $Y$ have? (Give the names and values of any parameters.)

108

An oocyte is a female germ cell involved in reproduction. Based on analyses of a large sample, the article “Reproductive Traits of Pioneer Gastropod Species Colonizing Deep-Sea Hydrothermal Vents After an Eruption” (Marine Biology, 2011: 181-192) proposed the following mixture of normal distributions as a model for the distribution of $X=$ oocyte diameter $\left(\mu \mathrm{m}\right)$ :

$$f\left(x\right)=p{f}_1\left(x;{\mu }_1,\sigma \right)+\left(1-p\right)f_2\left(x;{\mu }_2,\sigma \right)$$

where $f_1$ and $f_2$ are normal pdfs. Suggested parameter values were $p=.35,{\mu }_1=4.4,{\mu }_2=5.0$ , and $\sigma =.27$ .

a. What is the expected (i.e. mean) value of oocyte diameter?

b. What is the probability that oocyte diameter is between $4.4\mu \mathrm{m}$ and $5.0\mu \mathrm{m}$ ?

• Hint: Write an expression for the corresponding integral, carry the integral operation through to the two components, and then use the fact that each component is a normal pdf.

c. What is the probability that oocyte diameter is smaller than its mean value? What does this imply about the shape of the density curve?

109

The article “The Prediction of Corrosion by Statistical Analysis of Corrosion Profiles” (Corrosion Science, 1985: 305-315) suggests the following cdf for the depth $X$ of the deepest pit in an experiment involving the exposure of carbon manganese steel to acidified seawater.

$$F\left(x;\alpha ,\beta \right)=e^{-e^{-\left(x-\alpha \right)/\beta }}-\infty <x<\infty$$

The authors propose the values $\alpha =150$ and $\beta =90$ . Assume this to be the correct model.

a. What is the probability that the depth of the deepest pit is at most 150 ? At most 300 ? Between 150 and 300 ?

b. Below what value will the depth of the maximum pit be observed in $90\%$ of all such experiments?

c. What is the density function of $X$ ?

d. The density function can be shown to be unimodal (a single peak). Above what value on the measurement axis does this peak occur? (This value is the mode.)

e. It can be shown that $E\left(X\right)\approx .5772\beta +\alpha$ . What is the mean for the given values of $\alpha$ and $\beta$ , and how does it compare to the median and mode? Sketch the graph of the density function.

• Note: This is called the largest extreme value distribution.

110

Let $t=$ the amount of sales tax a retailer owes the government for a certain period. The article “Statistical Sampling in Tax Audits” (Statistics and the Law, 2008: 320-343) proposes modeling the uncertainty in $t$ by regarding it as a normally distributed random variable with mean value $\mu$ and standard deviation $\sigma$ (in the article, these two parameters are estimated from the results of a tax audit involving $n$ sampled transactions). If $a$ represents the amount the retailer is assessed, then an under-assessment results if $t>a$ and an over-assessment results if $a>t$ . The proposed penalty (i.e., loss) function for over- or under-assessment is $\mathrm{L}\left(a,t\right)=t-a$ if $t>a$ and $=k\left(a-t\right)$ if $t\le a(k>1$ is suggested to incorporate the idea that over-assessment is more serious than under-assessment).

a. Show that $a^{\ast }=\mu +\sigma {\Phi }^{-1}\left(1/\left(k+1\right)\right)$ is the value of $a$ that minimizes the expected loss, where ${\Phi }^{-1}$ is the inverse function of the standard normal cdf.

b. If $k=2$ (suggested in the article), \mu = \ {100},{000}$,and$\sigma = $ {10},{000}$,whatistheoptimalvalueof$a$ , and what is the resulting probability of over-assessment?

111

The mode of a continuous distribution is the value $x^{\ast }$ that maximizes $f\left(x\right)$ .

a. What is the mode of a normal distribution with parameters $\mu$ and $\sigma$ ?

b. Does the uniform distribution with parameters $A$ and $B$ have a single mode? Why or why not?

c. What is the mode of an exponential distribution with parameter $\lambda$ ? (Draw a picture.)

d. If $X$ has a gamma distribution with parameters $\alpha$ and $\beta$ , and $\alpha >1$ , find the mode.

• Hint: $\ln \left[f\left(x\right)\right]$ will be maximized iff $f\left(x\right)$ is, and it may be simpler to take the derivative of $\ln \left[f\left(x\right)\right]$ .

e. What is the mode of a chi-squared distribution having $\nu$ degrees of freedom?

112

The article “Error Distribution in Navigation” (J. of the Institute of Navigation, 1971: 429-442) suggests that the frequency distribution of positive errors (magnitudes of errors) is well approximated by an exponential distribution. Let $X=$ the lateral position error (nautical miles), which can be either negative or positive. Suppose the pdf of $X$ is

$$f\left(x\right)=\left(.1\right)e^{-.2|x|}-\infty <x<\infty$$

a. Sketch a graph of $f\left(x\right)$ and verify that $f\left(x\right)$ is a legitimate pdf (show that it integrates to 1).

b. Obtain the cdf of $X$ and sketch it.

c. Compute $P\left(X\le 0\right),P\left(X\le 2\right),P\left(-1\le X\le 2\right)$ , and the probability that an error of more than 2 miles is made.

113

The article “Statistical Behavior Modeling for Driver-Adaptive Precrash Systems” (IEEE Trans. on Intelligent Transp. Systems, 2013: 1-9) proposed the following mixture of two exponential distributions for modeling the behavior of what the authors called “the criticality level of a situation” $X$ .

$$f\left(x;{\lambda }_1,{\lambda }_2,p\right)=\{\begin{array}{cc}p{\lambda }_1{e}^{-{\lambda }_{1}x}+\left(1-p\right){\lambda }_2{e}^{-{\lambda }_{2}x}& x\ge 0\\ 0& \text{ otherwise }\end{array}$$

This is often called the hyperexponential or mixed exponential distribution. This distribution is also proposed as a model for rainfall amount in “Modeling Monsoon Affected Rainfall of Pakistan by Point Processes” (J. of Water Resources Planning and Mgmnt., 1992: 671-688).

a. Determine $E\left(X\right)$ and $V\left(X\right)$ . Hint: For $X$ distributed exponentially, $E\left(X\right)=1/\lambda$ and $V\left(X\right)=1/{\lambda }^2$ ; what does this imply about $E\left(X^2\right)$ ?

b. Determine the cdf of $X$ .

c. If $p=.5,{\lambda }_1=40$ , and ${\lambda }_2=200$ (values of the $\lambda$ ’s suggested in the cited article), calculate $P\left(X>.01\right)$ .

d. For the parameter values given in (c), what is the probability that $X$ is within one standard deviation of its mean value?

e. The coefficient of variation of a random variable (or distribution) is $CV=\sigma /\mu$ . What is $CV$ for an exponential rv? What can you say about the value of $CV$ when $X$ has a hyperexponential distribution?

f. What is $CV$ for an Erlang distribution with parameters $\lambda$ and $n$ as defined in Exercise 68?

• Note: In applied work, the sample $CV$ is used to decide which of the three distributions might be appropriate.

114

Suppose a particular state allows individuals filing tax returns to itemize deductions only if the total of all itemized deductions is at least \ {5000}$.Let$X$(in1000sofdollars)bethetotalofitemizeddeductionsonarandomlychosenform.Assumethat$X$ has the pdf

$$f\left(x;\alpha \right)=\{\begin{array}{cc}k/x^{\alpha }& x\ge 5\\ 0& \text{ otherwise }\end{array}$$

a. Find the value of $k$ . What restriction on $\alpha$ is necessary?

b. What is the cdf of $X$ ?

c. What is the expected total deduction on a randomly chosen form? What restriction on $\alpha$ is necessary for $E\left(X\right)$ to be finite?

d. Show that $\ln \left(X/5\right)$ has an exponential distribution with parameter $\alpha -1$ .

115

Let $I_i$ be the input current to a transistor and $I_0$ be the output current. Then the current gain is proportional to $\ln \left(I_0/I_i\right)$ . Suppose the constant of proportionality is 1

(which amounts to choosing a particular unit of measurement), so that current gain $=X=\ln \left(I_0/I_i\right)$ . Assume $X$ is normally distributed with $\mu =1$ and $\sigma =.05$ .

a. What type of distribution does the ratio $I_0/I_i$ have?

b. What is the probability that the output current is more than twice the input current?

c. What are the expected value and variance of the ratio of output to input current?

116

The article “Response of ${\mathrm{SiC}}_4/{\mathrm{Si}}_3{\mathrm{N}}_4$ Composites Under Static and Cyclic Loading-An Experimental and Statistical Analysis” (J. of Engr. Materials and Technology, 1997: 186-193) suggests that tensile strength (MPa) of composites under specified conditions can be modeled by a Weibull distribution with $\alpha =9$ and $\beta =180$ .

a. Sketch a graph of the density function.

b. What is the probability that the strength of a randomly selected specimen will exceed 175 ? Will be between 150 and 175 ?

c. If two randomly selected specimens are chosen and their strengths are independent of one another, what is the probability that at least one has a strength between 150 and 175 ?

d. What strength value separates the weakest $10\%$ of all specimens from the remaining $90\%$ ?

117

Let $\mathrm{Z}$ have a standard normal distribution and define a new rv $Y$ by $Y=\sigma Z+\mu$ . Show that $Y$ has a normal distribution with parameters $\mu$ and $\sigma$ .

• Hint: $Y\le y$ iff $Z\le$ ? Use this to find the cdf of $Y$ and then differentiate it with respect to $y$ .

118

a. Suppose the lifetime $X$ of a component, when measured in hours, has a gamma distribution with parameters $\alpha$ and $\beta$ . Let $Y=$ the lifetime measured in minutes. Derive the pdf of $Y$ .

• Hint: $Y\le y$ iff $X\le y/60$ . Use this to obtain the cdf of $Y$ and then differentiate to obtain the pdf.

b. If $X$ has a gamma distribution with parameters $\alpha$ and $\beta$ , what is the probability distribution of $Y=cX$ ?

119

In Exercises 117 and 118, as well as many other situations, one has the $\mathrm{pdf}f\left(x\right)$ of $X$ and wishes to know the pdf of $y=h\left(X\right)$ . Assume that $h\left(\cdot \right)$ is an invertible function, so that $y=h\left(x\right)$ can be solved for $x$ to yield $x=k\left(y\right)$ . Then it can be shown that the pdf of $Y$ is

$$g\left(y\right)=f\left[k\left(y\right)\right]\cdot \left|k^{\prime }\left(y\right)\right|$$

a. If $X$ has a uniform distribution with $A=0$ and $B=1$ , derive the pdf of $Y=-\ln \left(X\right)$ .

b. Work Exercise 117, using this result.

c. Work Exercise 118(b), using this result.

120

Based on data from a dart-throwing experiment, the article “Shooting Darts” (Chance, Summer 1997, 16-19) proposed that the horizontal and vertical errors from aiming at a point target should be independent of one another, each with a normal distribution having mean 0 and variance ${\sigma }^2$ . It can then be shown that the pdf of the distance $V$ from the target to the landing point is

$$f\left(v\right)=\frac{v}{{\sigma }^2}\cdot e^{-v^2/2{\sigma }^2}v>0$$

a. This pdf is a member of what family introduced in this chapter?

b. If $\sigma =20\mathrm{mm}$ (close to the value suggested in the paper), what is the probability that a dart will land within $25\mathrm{mm}$ (roughly 1 in.) of the target?

121

The article “Three Sisters Give Birth on the Same Day” (Chance, Spring 2001, 23-25) used the fact that three Utah sisters had all given birth on March 11, 1998 as a basis for posing some interesting questions regarding birth coincidences.

a. Disregarding leap year and assuming that the other 365 days are equally likely, what is the probability that three randomly selected births all occur on March 11? Be sure to indicate what, if any, extra assumptions you are making.

b. With the assumptions used in part (a), what is the probability that three randomly selected births all occur on the same day?

c. The author suggested that, based on extensive data, the length of gestation (time between conception and birth) could be modeled as having a normal distribution with mean value 280 days and standard deviation 19.88 days. The due dates for the three Utah sisters were March 15, April 1, and April 4, respectively. Assuming that all three due dates are at the mean of the distribution, what is the probability that all births occurred on March 11?

• Hint: The deviation of birth date from due date is normally distributed with mean 0.

d. Explain how you would use the information in part (c) to calculate the probability of a common birth date.

122

Let $X$ denote the lifetime of a component, with $f\left(x\right)$ and $F\left(x\right)$ the pdf and cdf of $X$ . The probability that the component fails in the interval $\left(x,x+\Delta x\right)$ is approximately $f\left(x\right)\cdot \Delta x$ . The conditional probability that it fails in $\left(x,x+\Delta x\right)$ given that it has lasted at least $x$ is $f\left(x\right)\cdot \Delta x/\left[1-F\left(x\right)\right]$ . Dividing this by $\Delta x$ produces the failure rate function:

$$r\left(x\right)=\frac{f\left(x\right)}{1-F\left(x\right)}$$

An increasing failure rate function indicates that older components are increasingly likely to wear out, whereas a decreasing failure rate is evidence of increasing reliability with age. In practice, a “bathtub-shaped” failure is often assumed.

a. If $X$ is exponentially distributed, what is $r\left(x\right)$ ?

b. If $X$ has a Weibull distribution with parameters $\alpha$ and $\beta$ , what is $r\left(x\right)$ ? For what parameter values will $r\left(x\right)$ be increasing? For what parameter values will $r\left(x\right)$ decrease with $x$ ?

c. Since $r\left(x\right)=-\left(d/dx\right)\ln \left[1-F\left(x\right)\right],\ln \left[1-F\left(x\right)\right]=$ $-\int r\left(x\right)dx$ . Suppose

$$r\left(x\right)=\{\begin{array}{cc}\alpha \left(1-\frac{x}{\beta }\right)& 0\le x\le \beta \\ 0& \text{ otherwise }\end{array}$$

so that if a component lasts $\beta$ hours, it will last forever (while seemingly unreasonable, this model can be used to study just “initial wearout”). What are the cdf and pdf of $X$ ?

123

Let $U$ have a uniform distribution on the interval $\left[0,1\right]$ . Then observed values having this distribution can be obtained from a computer’s random number generator. Let $X=-\left(1/\lambda \right)\ln \left(1-U\right)$ .

a. Show that $X$ has an exponential distribution with parameter $\lambda$ .

• Hint: The cdf of $X$ is $F\left(x\right)=P\left(X\le x\right)$ ; $X\le x$ is equivalent to $U\le$ ?

b. How would you use part (a) and a random number generator to obtain observed values from an exponential distribution with parameter $\lambda =10$ ?

124

Consider an $\mathrm{rv}X$ with mean $\mu$ and standard deviation $\sigma$ , and let $g\left(X\right)$ be a specified function of $X$ . The first-order Taylor series approximation to $g\left(X\right)$ in the neighborhood of $\mu$ is

$$g\left(X\right)\approx g\left(\mu \right)+g^{\prime }\left(\mu \right)\cdot \left(X-\mu \right)$$

The right-hand side of this equation is a linear function of $X$ . If the distribution of $X$ is concentrated in an interval over which $g\left(\cdot \right)$ is approximately linear

• e.g., $\sqrt{x}$ is approximately linear in $\left(1,2\right)]$ , then the equation yields approximations to $E\left(g\left(X\right)\right)$ and $V\left(g\left(X\right)\right)$ .

a. Give expressions for these approximations.

• Hint: Use rules of expected value and variance for a linear function $aX+b$ .

b. If the voltage $v$ across a medium is fixed but current $I$ is random, then resistance will also be a random variable related to $I$ by $R=v/I$ . If ${\mu }_I=20$ and ${\sigma }_I=.5$ , calculate approximations to ${\mu }_R$ and ${\sigma }_R$ .

125

A function $g\left(x\right)$ is convex if the chord connecting any two points on the function’s graph lies above the graph. When $g\left(x\right)$ is differentiable, an equivalent condition is that for every $x$ , the tangent line at $x$ lies entirely on or below the graph. (See the figure below.) How does $g\left(\mu \right)=g\left(E\left(X\right)\right)$ compare to $E\left(g\left(X\right)\right)$ ?

• Hint: The equation of the tangent line at $x=\mu$ is $y=g\left(\mu \right)+g^{\prime }\left(\mu \right)\cdot \left(x-\mu \right)$ . Use the condition of convexity, substitute $X$ for $x$ , and take expected values.
• Note: Unless $g\left(x\right)$ is linear, the resulting inequality (usually called Jensen’s inequality) is strict $\left(<\text{rather than}\le \right)$ ; it is valid for both continuous and discrete rv’s.

126

Let $X$ have a Weibull distribution with parameters $\alpha =2$ and $\beta$ . Show that $Y=2X^2/{\beta }^2$ has a chi-squared distribution with $\nu =2$ .

• Hint: The cdf of $Y$ is $P\left(Y\le y\right)$ ; express this probability in the form $P\left(X\le g\left(y\right)\right)$ , use the fact that $X$ has a cdf of the form in Expression (4.12), and differentiate with respect to $y$ to obtain the pdf of $Y$ .

127

An individual’s credit score is a number calculated based on that person’s credit history that helps a lender determine how much he/she should be loaned or what credit limit should be established for a credit card. An article in the Los Angeles Times gave data which suggested that a beta distribution with parameters $A=150,B=850$ , $\alpha =8,\beta =2$ would provide a reasonable approximation to the distribution of American credit scores.

• Note: credit scores are integer-valued

a. Let $X$ represent a randomly selected American credit score. What are the mean value and standard deviation of this random variable? What is the probability that $X$ is within 1 standard deviation of its mean value?

b. What is the approximate probability that a randomly selected score will exceed 750 (which lenders consider a very good score)?

128

Let $V$ denote rainfall volume and $W$ denote runoff volume (both in mm). According to the article “Runoff Quality Analysis of Urban Catchments with Analytical Probability Models” (J. of Water Resource Planning and Management, 2006: 4-14), the runoff volume will be 0 if $V\le v_d$ and will be $k\left(V-v_d\right)$ if $V>v_d$ . Here $v_d$ is the volume of depression storage (a constant), and $k$ (also a constant) is the runoff coefficient. The cited article proposes an exponential distribution with parameter $\lambda$ for $V$ .

a. Obtain an expression for the cdf of $W$ .

• Note: $W$ is neither purely continuous nor purely discrete; instead it has a “mixed” distribution with a discrete component at 0 and is continuous for values $w>0$ .

b. What is the pdf of $W$ for $w>0$ ? Use this to obtain an expression for the expected value of runoff volume.