91Ó°ÊÓ

Let X denote the time to failure (in years) of a certain hydraulic component. Suppose the pdf of X is \({\bf{f}}\left( {\bf{x}} \right) = {\bf{32}}/{\left( {{\bf{x}} + {\bf{4}}} \right)^{\bf{3}}}{\rm{ }}{\bf{for}}{\rm{ }}{\bf{x}} < {\bf{0}}\). a. Verify that f (x) is a legitimate pdf. b. Determine the cdf. c. Use the result of part (b) to calculate the probability that the time to failure is between \({\bf{2}}{\rm{ }}{\bf{and}}{\rm{ }}{\bf{5}}\)years. d. What is the expected time to failure? e. If the component has a salvage value equal to \({\bf{100}}/\left( {{\bf{4}} + {\bf{x}}} \right)\)when it is time to fail is x, what is the expected salvage value?

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

\(\left\{ {\begin{array}{*{20}{c}}{0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x < 0}\\{\frac{{{x^3}}}{3}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0 \le x \le \frac{7}{3}}\\{1 - \frac{1}{2}\left( {\frac{7}{3} - x} \right)\left( {\frac{7}{4} - \frac{3}{4}x} \right)\,\,\,\,\,\,1 \le x \le \frac{7}{3}}\\{1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x > \frac{7}{3}}\end{array}} \right.\)

a. Obtain the pdf f (x) and sketch its graph.

b. Compute\({\bf{P}}\left( {.{\bf{5}} \le {\bf{X}} \le {\bf{2}}} \right)\). c. Compute E(X).

When circuit boards used in the manufacture of compact disc players are tested, the long-run percentage of defectives is\({\rm{5\% }}\). Suppose that a batch of \({\rm{250}}\) 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 \({\rm{10\% }}\) of the boards in the batch are defective?

b. What is the approximate probability that there are exactly \({\rm{10}}\) defectives in the batch?

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

\(f(x)= \left\{ {\begin{array}{*{20}{c}}{\frac{3}{2} \times \frac{1}{{{x^2}}}}&{1£x£3} \\0&{{\text{ }}otherwise{\text{ }}}\end{array}} \right.\)

a. Obtain the cdf.

b. What is the probability that reaction time is at most\({\rm{2}}{\rm{.5sec}}\)? Between \({\rm{1}}{\rm{.5}}\) and\({\rm{2}}{\rm{.5sec}}\)?

c. Compute the expected reaction time.

d. Compute the standard deviation of reaction time.

e. If an individual takes more than \({\rm{1}}{\rm{.5sec}}\) 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 \({\rm{h(X) = }}\) the time that the light is on as a function of reaction time\({\rm{X}}\).)

Let \({\rm{X}}\) denote the temperature at which a certain chemical reaction takes place. Suppose that \({\rm{X}}\) has pdf

\(f(x) = \left\{ {\begin{aligned}{{}{}}{\frac{1}{9}\left( {4 - {x^2}} \right)}&{ - 1£x£2} \\0&{{\text{ }}otherwise{\text{ }}}\end{aligned}} \right.\)

a. Sketch the graph of \({\rm{f(x)}}\).

b. Determine the cdf and sketch it.

c. Is \({\rm{0}}\) the median temperature at which the reaction takes place? If not, is the median temperature smaller or larger than\({\rm{0}}\)?

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 \({\rm{Y = }}\) the number among the ten labs at which the temperature exceeds \({\rm{1}}\). What kind of distribution does \({\rm{Y}}\) have? (Give the names and values of any parameters.)

A family of pdf’s that has been used to approximate the distribution of income, city population size, and size of firms is the Pareto family. The family has two parameters, \({\rm{k}}\) and \({\rm{\theta }}\), both\({\rm{ > 0}}\), and the pdf is

\({\rm{f(x;\theta ) = \{ }}\begin{array}{*{20}{c}}{\frac{{{\rm{k}} \cdot {{\rm{\theta }}^{\rm{k}}}}}{{{{\rm{x}}^{{\rm{k + 1}}}}}}}&{{\rm{x}} \ge {\rm{\theta }}}\\{\rm{0}}&{{\rm{x < \theta }}}\end{array}\)

a. Sketch the graph of \({\rm{f(x;\theta )}}\).

b. Verify that the total area under the graph equals \({\rm{1}}\).

c. If the rv \({\rm{X}}\) has pdf \({\rm{f(x;\theta )}}\), for any fixed \({\rm{b > \theta }}\), obtain an expression for \({\rm{P(X}} \le {\rm{b)}}\).

d. For \({\rm{\theta < a < b}}\) obtain an expression for the probability \({\rm{P(a}} \le {\rm{X}} \le {\rm{b)}}\).

The mode of a continuous distribution is the value \({{\rm{x}}^{\rm{*}}}\) that maximizes\({\rm{f(x)}}\).

a. What is the mode of a normal distribution with parameters \({\rm{\mu }}\) and \({\rm{\sigma }}\) ?

b. Does the uniform distribution with parameters \({\rm{A}}\) and \({\rm{B}}\) have a single mode? Why or why not?

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

d. If \({\rm{X}}\) has a gamma distribution with parameters \({\rm{\alpha }}\) and\({\rm{\beta }}\), and\({\rm{\alpha > 1}}\), find the mode. (Hint: \({\rm{ln(f(x))}}\)will be maximized if \({\rm{f(x)}}\) is, and it may be simpler to take the derivative of\({\rm{ln(f(x))}}\).)

e. What is the mode of a chi-squared distribution having \({\rm{v}}\) degrees of freedom?

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

\(f(x) = (.1){e^{ - .2k1}} - ¥ < x < ¥\)

a. Sketch a graph of \({\rm{f(x)}}\) and verify that \({\rm{f(x)}}\) is a legitimate pdf (show that it integrates to\({\rm{1}}\)).

b. Obtain the cdf of \({\rm{X}}\) and sketch it.

c. Compute \(P(X£0),P(X£2),P( - 1£X£2)\), and the probability that an error of more than \({\rm{2}}\) miles is made.

The article "'Response of \({\rm{Si}}{{\rm{C}}_{\rm{i}}}{\rm{/S}}{{\rm{i}}_{\rm{3}}}{\rm{\;}}{{\rm{N}}_{\rm{4}}}\) Composites Under Static and Cyclic Loading-An Experimental and Statistical Analysis" (J. of Engr. Materials and Technology, \({\rm{1997: 186 - 193}}\)) suggests that tensile strength (MPa) of composites under specified conditions can be modeled by a Weibull distribution with \({\rm{\alpha = 9}}\) and\({\rm{\beta = 180}}\).

a. Sketch a graph of the density function.

b. What is the probability that the strength of a randomly selected specimen will exceed \({\rm{175}}\)? Will be between \({\rm{150}}\) and \({\rm{175}}\)?

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 \({\rm{150}}\) and\({\rm{175}}\)?

d. What strength value separates the weakest \({\rm{10\% }}\) of all specimens from the remaining\({\rm{90\% }}\)?

Example \({\rm{4}}{\rm{.5}}\) introduced the concept of time headway in traffic flow and proposed a particular distribution for \({\rm{X = }}\) the headway between two randomly selected consecutive cars (sec). Suppose that in a different traffic environment, the distribution of time headway has the form

\({\rm{f(x) = }}\left\{ {\begin{array}{*{20}{c}}{\frac{{\rm{k}}}{{{{\rm{x}}^{\rm{4}}}}}}&{{\rm{x > 1}}}\\{\rm{0}}&{{\rm{x}} \le {\rm{1}}}\end{array}} \right.\)