Questions and Answers of Introduction To Probability Statistics

With reference to Exercise 4.95, find the percent of the time that the interval between breakdowns of the computer will be(a) less than 1 week;(b) at least 5 weeks.Data From Exercise 4.95 4.95 The
With reference to Exercise 4.58, find the probabilities that the time between successive requests for consulting will be(a) less than 0.5 week;(b) more than 3 weeks.Data From Exercise 4.58 4.58 A
Given a Poisson process with on the average \(\alpha\) arrivals per unit time, find the probability that there will be no arrivals during a time interval of length \(t\), namely, the probability that
Use the result of Exercise 5.61 to find an expression for the probability density of the waiting time between successive arrivals.Data From Exercise 5.61 5.61 Given a Poisson process with on the
Verify for \(\alpha=3\) and \(\beta=3\) that the integral of the beta density, from 0 to 1 , is equal to 1 .
If the ratio of defective switches produced during complete production cycles in the previous month can be looked upon as a random variable having a beta distribution with \(\alpha=3\) and
Suppose the proportion of error in code developed by a programmer, which varies from software to software, may be looked upon as a random variable having the beta distribution with \(\alpha=2\) and
Show that when \(\alpha>1\) and \(\beta>1\), the beta density has a relative maximum at\[x=\frac{\alpha-1}{\alpha+\beta-2}\]
With reference to the Example 19, find the probability that such a battery will not last 100 hours.Data From Example 19 EXAMPLE 19 A large sample test of the mean amount of cheese Refer to Example 8,
Suppose that the time to failure (in minutes) of certain electronic components subjected to continuous vibrations may be looked upon as a random variable having the Weibull distribution with
Suppose that the processing speed (in milliseconds) of a supercomputer is a random variable having the Weibull distribution with \(\alpha=0.005\) and \(\beta=0.125\). What is the probability that
Verify the formula for the variance of the Weibull distribution given on page 160. Mean of Weibull distribution Variance of Weibull distribution + Using a similar method to determine first 2, the
Two transistors are needed for an integrated circuit. Of the eight available, three have broken insulation layers, two have poor diodes, and three are in good condition. Two transistors are selected
Two random variables are independent and each has a binomial distribution with success probability 0.7 and 4 trials.(a) Find the joint probability distribution.(b) Find the probability that the first
If two random variables have the joint density\[f\left(x_{1}, x_{2}\right)= \begin{cases}x_{1} x_{2} & \text { for } 0
With reference to Exercise 5.73, find the joint cumulative distribution function of the two random variables, the cumulative distribution functions of the individual random variables, and check
If two random variables have the joint density \(f(x, y)= \begin{cases}\frac{6}{5}\left(x+y^{2}\right) & \text { for } 0
With reference to Exercise 5.76, find both marginal densities and use them to find the probabilities that(a) \(X>0.8\);(b) \(YData From Exercise 5.76 5.76 If two random variables have the joint
With reference to Exercise 5.76, find(a) an expression for \(f_{1}(x \mid y)\) for \(0(b) an expression for \(f_{1}(x \mid 0.5)\);(c) the mean of the conditional density of the first random variable
With reference to Example 27, find expressions for(a) the conditional density of the first random variable when the second takes on the value \(x_{2}=0.25\);(b) the conditional density of the second
If three random variables have the joint density\[f(x, y, z)=\left\{\begin{array}{lc}k(x+y) e^{-z} & \text { for } 0
A precision drill positioned over a target point will make an acceptable hole if it is within 5 microns of the target. Using the target as the origin of a rectangular system of coordinates, assume
with \(\mu_{1}=\mu_{2}=0\) and \(\sigma=2\). What is the probability that the hole will be acceptable?
With reference to Exercise 5.73, find the expected value of the random variable whose values are given by \(g\left(x_{1}, x_{2}\right)=x_{1}+x_{2}\).Data From Exercise 5.73 5.73 If two random
With reference to Exercise 5.76, find the expected value of the random variable whose values are given by \(g(x, y)=x^{2} y\).Data From Exercise 5.76 5.76 If two random variables have the joint
If measurements of the length and the width of a rectangle have the joint density\[f(x, y)=\left\{\begin{array}{cc}\frac{1}{a b} & \text { for } L-\frac{a}{2}
Establish a relationship between \(f_{1}\left(x_{1} \mid x_{2}\right)\), \(f_{2}\left(x_{2} \mid x_{1}\right), f_{1}\left(x_{1}\right)\), and \(f_{2}\left(x_{2}\right)\).
If \(X_{1}\) has mean 1 and variance 5 while \(X_{2}\) has mean - 1 and variance 5 , and the two are independent, find(a) \(E\left(X_{1}+X_{2}\right)\);(b)
If \(X_{1}\) has mean 8 and variance 2 while \(X_{2}\) has mean -12.5 and variance 2.25 , and the two are independent, find(a) \(E\left(X_{1}-X_{2}\right)\);(b)
If \(X_{1}\) has mean 1 and variance 3 while \(X_{2}\) has mean -2 and variance 5 , and the two are independent, find(a) \(E\left(X_{1}+2 X_{2}-3\right)\)(b) \(\operatorname{Var}\left(X_{1}+2
The time taken by a traditional nuclear reactor to generate one nuclear chain reaction with fast neutrons, \(X_{1}\), has mean 10 nanoseconds and variance 4 , while the time taken by an improved
Let \(X_{1}, X_{2}, \ldots, X_{20}\) be independent and let each have the same marginal distribution with mean 10 and variance 3. Find(a) \(E\left(X_{1}+X_{2}+\cdots+X_{20}\right)\);(b)
Let \(f(x)=0.2\) for \(x=0,1,2,3,4\).(a) Find the moment generating function.(b) Obtain \(E(X)\) and \(E\left(X^{2}\right)\) by differentiating the moment generating function.
Let\[f(x)=0.40\left(\begin{array}{l}4 \\x\end{array}\right) \quad \text { for } x=0,1,2,3,4\](a) Find the moment generating function.(b) Obtain \(E(X)\) and \(E\left(X^{2}\right)\) by differentiating
Let \(Z\) have a normal distribution with mean 0 and variance 1 .(a) Find the moment generating function of \(Z^{2}\).(b) Identify the distribution of \(Z^{2}\) by recognizing the form of the moment
Let \(X\) be a continuous random variable having probability density function\[f(x)= \begin{cases}2 e^{-2 x} & \text { for } x>0 \\ 0 & \text { elsewhere }\end{cases}\](a) Find the moment generating
Establish the result in Example 41 concerning the difference of two independent normal random variables, \(X\) and \(Y\).Data From Example 41 EXAMPLE 41 Sum of two independent normal random variables
Let \(X\) and \(Y\) be independent normal random variables with\[\begin{array}{lll}E(X)=4 & \text { and } & \sigma_{X}^{2}=25 \\E(Y)=3 & \text { and } & \sigma_{Y}^{2}=16\end{array}\](a) Use moment
Let \(X\) have the geometric distribution\[f(x)=p(1-p)^{x-1} \quad \text { for } x=1,2, \ldots\](a) Obtain the moment generating function for\[t
For any 11 observations,(a) Use software or Table 3 to verify the normal scores \(-1.38-0.97-0.67-0.43-0.21 \quad 0 \quad 0.210 .430 .670 .971 .38\)(b) Construct a normal scores plot using the
(Transformations) The MINITAB commands Dialog box: Calc Calculator Type C2 in Store. Type LOGE(C1) in Expression. Click OK. Calc Calculator Type C3 in Store. Type SQRT(C1) Expression. Click OK. Calc
Verify that(a) the exponential density \(0.3 e^{-0.3 x}, x>0\) corresponds to the distribution function \(F(x)=\) \(1-e^{-0.3 x}, x>0\)(b) the solution of \(u=F(x)\) is given by \(x=\) \([-\ln (1-u)]
Verify that(a) the Weibull density \(\alpha \beta x^{\beta-1} e^{-a x^{\beta}}, x>0\), corresponds to the distribution function \(F(x)=\) \(1-e^{-a x^{\beta}}, x>0\)(b) the solution of \(u=F(x)\) is
Consider two independent standard normal variables whose joint probability density is\[\frac{1}{2 \pi} e^{-\left(z_{1}^{2}+z_{2}^{2}\right) / 2}\]Under a change to polar coordinates, \(z_{1}=\) \(r
The statistical package MINITAB has a random number generator. To simulate 5 values from an exponential distribution having mean \(\beta=0.05\), chooseOutput:One call produced the output
The statistical package MINITAB has a normal random number generator. To simulate 5 values from a normal distribution having mean 7 and standard deviation 4, and place them in \(\mathrm{C}\), use the
If the probability density of a random variable is given by\[f(x)= \begin{cases}k\left(1-x^{2}\right) & \text { for } 0
In certain experiments, the error made in determining the density of a silicon compound is a random variable having the probability density\[f(x)=\left\{\begin{aligned}25 & \text { for }-0.02
A coil is rotated in a magnetic field to generate current. The voltage generated can be modeled by a normal distribution having mean \(\mu\) and standard deviation \(0.5 \mathrm{~V}\) where \(\mu\)
Referring to Exercise 5.112, suppose the rotation speed of the coil can be increased and standard deviation decreased. Determine the new value for the standard deviation that would restrict the
The burning time of an experimental rocket is a random variable having the normal distribution with \(\mu=4.76\) seconds and \(\sigma=0.04\) second. What is the probability that this kind of rocket
Verify that(a) \(z_{0.10}=1.28\)(b) \(z_{0.001}=3.09\).
Referring to Exercise 5.28, find the quartiles of the normal distribution with \(\mu=102\) and \(\sigma=27\).Data From Exercise 5.28 5.28 Find the quartiles -20.25 20.50 20.25 of the standard normal
The probability density shown in Figure 5.19 is the log-normal distribution with \(\alpha=8.85\) and \(\beta=1.03\). Find the probability that(a) the inter request time is more than 200
The probability density shown in Figure 5.21 is the exponential distribution\[f(x)= \begin{cases}0.55 e^{-0.55 x} & 0Find the probability that(a) the time to observe a particle is more than 200
Referring to the normal scores in Exercise 5.101, construct a normal scores plot of the current flow data in Exercise 2.68.Data From Exercise 5.101Data From Exercise 2.68 5.101 For any 11
A change is made to one product page on the retail companies' web site. To determine if the change does improve the efficiency of that product page, data must be collected on the proportion of
If \(n\) salespeople are employed in a door-to-door selling campaign, the gross sales volume in thousands of dollars may be regarded as a random variable having the gamma distribution with
A software engineer models the crashes encountered when executing a new software as a random variable having the Weibull distribution with \(\alpha=0.06\) and \(\beta=6.0\). What is the probability
Let the times to breakdown for the processors of a parallel processing machine have joint density\[f(x, y)= \begin{cases}0.04 e^{-0.2 x-0.2 y} & \text { for } x>0, y>0 \\ 0 & \text { elsewhere
Two random variables are independent and each has a binomial distribution with success probability 0.6 and 2 trials.(a) Find the joint probability distribution.(b) Find the probability that the
If \(X_{1}\) has mean -5 and variance 3 while \(X_{2}\) has mean 1 and variance 4 , and the two are independent, find(a) \(E\left(3 X_{1}+5 X_{2}+2\right)\);(b) \(\operatorname{Var}\left(3 X_{1}+5
Let \(X_{1}, X_{2}, \ldots, X_{50}\) be independent and let each have the same marginal distribution with mean -5 and variance 8 . Find(a) \(E\left(X_{1}+X_{2}+\cdots+X_{50}\right)\);(b)
Refer to Example 7 concerning scanners. The maximum attenuation has a normal distribution with mean \(10.1 \mathrm{~dB}\) and standard deviation \(2.7 \mathrm{~dB}\).(a) What proportion of the
Find the mean and variance of the binomial distribution with \(n=6\) and \(p=0.55\) by using(a) Table 1 and the formulas defining \(\mu\) and \(\sigma^{2}\);(b) The special formulas for the mean and
Construct a table showing the upper limits provided by Chebyshev's theorem for the probabilities of obtaining values differing from the mean by at least 1 , 2 , and 3 standard deviations and also the
Use the recursion formula of Exercise 4.50 to calculate the value of the Poisson distribution with \(\lambda=3\) for \(x=0,1,2, \ldots\), and 9 , and draw the probability histogram of this
Use Table 2W or software to find(a) \(F(4 ; 7)\);(b) \(f(4 ; 7)\);(c) \(\sum_{k=6}^{19} f(k ; 8)\). Number of radio messages 0 Observed frequencies Poisson probabilities Expected frequencies 3 0.010
Use Table 2W or software to find(a) \(F(9 ; 12)\);(b) \(f(9 ; 12)\);(c) \(\sum_{k=3}^{12} f(k ; 7.5)\). Number of radio messages Observed frequencies Poisson probabilities Expected frequencies
Use the Poisson distribution to approximate the binomial probability b(3 ; 100,0.03).
In a "torture test," a light switch is turned on and off until it fails. If the probability that the switch will fail any time it is turned on or off is 0.001, what is the probability that the switch
Use the formulas defining \(\mu\) and \(\sigma^{2}\) to show that the mean and the variance of the Poisson distribution are both equal to \(\lambda\).
The formula of Exercise 3.52 is often used to determine subjective probabilities. For instance, if an applicant for a job “feels” that the odds are 7 to 4 of getting the job, the subjective
Refer to Example 31 concerning spam but now suppose that among the 5000 messages, the 1750 spam messages have 1570 that contain the words on a new list and that the 3250 normal messages have 300 that
Refer to Example 12 of motors for miniaturized capsules, but instead suppose that 20 motors are available and that 4 will not operate satisfactorily, when placed in a capsule. If the scientist wishes
Damages at a factory manufacturing chairs are categorized according to the material wasted.plastic75iron31cloth22spares8Draw a Pareto chart.
The following are figures on sacks of cement used daily at a construction site: 75,77,82,45,55,90,80, 81,76,47,59,52,71,83,91,76,57,59,43 and 79. Construct a stem-and-leaf display with the stem
The following are determinations of a river's annual maximum flow in cubic meters per second: 405,355,419,267,370,391,612,383,434,462,288,317,540, 295, and 508. Construct a stem-and-leaf display with
The Aerokopter AK1-3 is an ultra-lightweight manned kit helicopter with a high rotor tip speed. A sample of 8 measurements of speed, in meters per second, yielded204 208 205 211
Find the mean and the standard deviation of the 20 humidity readings on page 31 by using the(a) the raw (ungrouped) data(b) the distribution obtained in that example Humidity Readings 10-19 20-29
With reference to the aluminum-alloy strength data in Example 7, make a stem-and-leaf display.Data From Example 7 EXAMPLE 7 A density histogram has total area I Compressive strength was measured on
Suppose that, next month, the quality control division will inspect 30 units. Among these, 20 will undergo a speed test and 10 will be tested for current flow. If an engineer is randomly assigned 4
A maker of specialized instruments receives shipments of 24 circuit boards. Suppose one shipment contains 4 that are defective. An engineer selects a random sample of size 4 . What are the
If 6 of 18 new buildings in a city violate the building code, what is the probability that a building inspector, who randomly selects 4 of the new buildings for inspection, will catch(a) none of the
Among the 13 countries that an international trade federation is considering for their next 4 annual conferences, 6 are in Asia. To avoid arguments, the selection is left to chance. If none of the
A shipment of 120 burglar alarms contains 5 that are defective. If 3 of these alarms are randomly selected and shipped to a customer, find the probability that the customer will get one bad unit by
Refer to Exercise 4.24 but now suppose there will be 75 units among which 45 will need to undergo a speed test and 30 will be tested for current flow. Find the probability that, among the four
Binomial probabilities can be calculated using MINITAB.Output:Probability Density Function Binomial with \(n=7\) and \(p=0.33\)\[\begin{array}{rr}x & P(X=x) \\2 & 0.308760 \end{array}\]Find
Cumulative binomial probabilities can be calculated using MINITAB.Output:Cumulative Distribution Function Binomial with \(n=7\) and \(p=0.33\)\[\begin{array}{lr}x & P(X2 & 0.578326
Suppose that the probabilities are \(0.4,0.3,0.2\), and 0.1 that there will be \(0,1,2\), or 3 power failures in a certain city during the month of July. Use the formulas which define \(\mu\) and
The following table gives the probabilities that a certain computer will malfunction \(0,1,2,3,4,5\), or 6 times on any one day:Use the formulas which define \(\mu\) and \(\sigma\) to find(a) the
Find the mean and the variance of the uniform probability distribution given by\[f(x)=\frac{1}{n} \quad \text { for } x=1,2,3, \ldots, n\]The sum of the first \(n\) positive integers is \(n(n+1) /
As can easily be verified by means of the formula for the binomial distribution (or by listing all 16 possibilities), the probabilities of getting \(0,1,2,3\), or 4 red cards in four draws from a
With reference to Exercise 4.38, find the variance of the probability distribution using(a) the formula that defines \(\sigma^{2}\);(b) the computing formula for \(\sigma^{2}\);(c) the special
If \(95 \%\) of certain high-performance radial tires last at least 30,000 miles, find the mean and the standard deviation of the distribution of the number of these tires, among 20 selected at
Find the mean and the standard deviation of the distribution of each of the following random variables (having binomial distributions):(a) The number of heads obtained in 676 flips of a balanced
Find the mean and the standard deviation of the hypergeometric distribution with the parameters \(n=3\), \(a=4\), and \(N=8\)(a) by first calculating the necessary probabilities and then using the
Prove the formula for the mean of the hypergeometric distribution with the parameters \(n, a\), and \(N\), namely, \(\mu=n \cdot \frac{a}{N}\).\[\sum_{r=0}^{k}\left(\begin{array}{c}m
Over the range of cylindrical parts manufactured on a computer-controlled lathe, the standard deviation of the diameters is 0.002 millimeter.(a) What does Chebyshev's theorem tell us about the
In 1 out of 22 cases, the plastic used in microwave friendly containers fails to meet heat standards. If 979 specimens are tested, what does Chebyshev's theorem tell us about the probability of

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