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introduction to probability statistics

Questions and Answers of Introduction To Probability Statistics

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
Show that for 48 million draws from a fair deck of cards, the probability is at least 0.9375 that the proportion of spades drawn will fall between 0.24975 and 0.25025 .
The time taken by students to fill out a loan request form has standard deviation 1.2 hours. What does Chebyshev’s theorem tell us about the probability that a students’ time will be within 4
Prove that (a) = E(X2)-; (b) 33 +2.
Prove that for the Poisson distribution\[\frac{f(x+1 ; \lambda)}{f(x ; \lambda)}=\frac{\lambda}{x+1}\]for \(x=0,1,2, \ldots\)
In a factory, \(8 \%\) of all machines break down at least once a year. Use the Poisson approximation to the binomial distribution to determine the probabilities that among 25 machines (randomly
During inspection of the continuous process of making large rolls of floor coverings, 0.5 imperfections are spotted per minute on average. Use the Poison distribution to find the probabilities(a) one
The number of gamma rays emitted per second by a certain radioactive substance is a random variable having the Poisson distribution with \(\lambda=5.8\). If a recording instrument becomes inoperative
A consulting engineer receives, on average, 0.7 requests per week. If the number of requests follows a Poisson process, find the probability that(a) in a given week, there will be at least 1
A conveyor belt conveys finished products to the warehouse at an average of 2 per minute. Find the probabilities that(a) at most 3 will be conveyed in a given minute;(b) at least 2 will be conveyed
Environmental engineers, concerned about the effects of releasing warm water from a power plants' cooling system into a Great Lake, decided to sample many organisms both inside and outside of a warm
An automated weight monitor can detect underfilled cans of beverages with probability 0.98. What is the probability it fails to detect an underfilled can for the first time when it encounters the
A company manufactures hydraulic gears, and records show that the probability is 0.04 that one of its new gears will fail its inspection test. What is the probability that the fifth gear in a day
Referring to Exercise 4.63, find the probability that the 15th gear in a day is the fourth to fail the test.Data From Exercise 4.63 4.63 A company manufactures hydraulic gears, and records show that
During an assembly process, parts arrive just as they are needed. However, at one station, the probability is 0.01 that a defective part will arrive in a one-hour period. Find the probability that(a)
The arrival of trucks at a receiving dock is a Poisson process with a mean arrival rate of 2 per hour.(a) Find the probability that exactly 5 trucks arrive in a two-hour period.(b) Find the
The number of flaws in a fiber optic cable follows a Poisson process with an average of 0.6 per 100 feet.(a) Find the probability of exactly 2 flaws in a 200 foot cable.(b) Find the probability of
Differentiating with respect to \(p\) on both sides of the equation\[\sum_{x=1}^{\infty} p(1-p)^{x-1}=1\]show that the geometric distribution\[f(x)=p(1-p)^{x-1} \quad \text { for } x=1,2,3,
Poisson probabilities can be calculated using MINITAB.Output:Poisson with mean \(=1.64\)Find the Poisson probabilities for \(x=2\) and \(x=3\) when(a) \(\lambda=2.73\);(b) \(\lambda=4.33\). Dialog
A civil engineer suspects that the cement being supplied for constructing a dam is adulterated. There are three units of white cement and three units of black cement. He will check all six for
With reference to Exercise 3.1, which of the three pairs of events, \(A\) and \(B, B\) and \(C\), and \(B\) and \(C\), are mutually exclusive?Data From Exercise 3.1 3.1 A civil engineer suspects that
With reference to Exercise 3.1, list the outcomes comprising each of the following events, and also express the events in words.(a) \(A \cup B\)(b) \(B \cap C\)(c) \(\bar{B}\)Data From Exercise 3.1
With reference to the sample space of Figure 3.1, express each of the following events in words.(a) \(F=\{(1,0),(1,1)\}\)(b) \(G=\{(0,2),(1,1),(2,0)\}\)(c) \(F \cap G\)Data From Figure 3.1 Figure
To construct sample spaces for experiments in which we deal with non-numerical data, we often code the various alternatives by assigning them numbers. For instance, if an engineer is asked to rate
With reference to Exercise 3.5, which of the pairs of events, \(P\) and \(Q, Q\) and \(R, R\) and \(S\), and \(P\) and \(S\), are mutually exclusive?Data From Exercise 3.5 3.5 To construct sample
Four supervisors and 3 engineers are responsible for work at a construction site, and at least 2 supervisors and one engineer have to be present at all times.(a) Using two coordinates so that
For each of the following experiments, decide whether it would be appropriate to use a sample space which is finite, countably infinite, or continuous.(a) A Geiger counter, located adjacent to a
With reference to Exercise 3.9, what events are represented by(a) regions 1 and 3 together;(b) regions 3 and 4 together;(c) regions 1,2, and 3 together?Data From Exercise 3.9 3.9 In Figure 3.6, C
With reference to Figure 3.4, what events are represented by(a) region 5;(b) regions 4 and 6 together;(c) regions 7 and 8 together;(d) regions 1, 2, 3 and 5 together?Data From Figure 3.4 Figure 3.4
With reference to Figure 3.4, what regions or combinations of regions represent the events that a motor will have(a) none of the major defects;(b) a shaft that is large and windings improper;(c) a
Use Venn diagrams to verify that(a) \(\overline{A \cup B}=\bar{A} \cap \bar{B}\)(b) \(B \cap(A \cup B)=B\)(c) \((A \cup B) \cap(\bar{A} \cup B)=B\)(d) \(A \cap B=(A \cup B) \cap(\bar{A} \cup B)
A building inspector has to check the wiring in a new apartment building either on Monday, Tuesday, Wednesday, or Thursday, and at 8 A.M., 1 P.M., or 2 P.M. A tree diagram which shows the various
If the five finalists in an international volleyball tournament are Spain, the United States, Uruguay, Portugal, and Japan, draw a tree diagram that shows the various possible first- and second-place
If a number cannot be immediately repeated, how many different three number combinations are possible for a combination lock with numbers \(0,1, \ldots, 29\).
Students are offered three cooperative training programs at local companies and four training programs outside the state. Count the number of possible training opportunities if an opportunity
You are required to choose a four digit personal identification number (PIN) for a new debit card. Each digit is selected from \(0,1, \ldots, 9\). How many choices do you have.
An Engineers Association consists of 5 civil engineers and 5 mechanical engineers.(a) In how many ways can a committee of 3 civil engineers and 2 mechanical engineers be appointed?(b) If 2 civil
If there are 9 cars in a race, in how many different ways can they place first, second, and third?
In how many ordered ways can a television director schedule 6 different commercials during the 6 time slots allocated to commercials during the telecast of the first period of a hockey game?
Last year; the maximum daily temperature in a plants' server room exceeded \(68^{\circ} \mathrm{F}\) in 12 days. Estimate the probability that the maximum temperature will exceed \(68^{\circ}
Among 150 persons interviewed as part of an urban mass transportation study, some live more than 3 miles from the center of the city \((A)\), some now regularly drive their own car to work \((B)\),
Refer to parts(d) and(c) of Exercise 3.13 to show that(a) \(P(A \cap B) \leq P(A)\);(b) \(P(A \cup B) \geq P(A)\).Data From Exercise 3.13 3.13 Use Venn diagrams to verify that (a) AUB AnB (b) BN
Explain why there must be a mistake in each of the following statements:(a) The probability that a student will get an \(A\) in a geology course is 0.3, and the probability that he or she will get
Use the definition of Exercise 3.51 to show that if the odds for the occurrence of event A are a to b, where a and b are positive integers, thenData From Exercise 3.51 P = a a+b
Subjective probabilities may or may not satisfy the third axiom of probability. When they do, we say that they are consistent; when they do not, they ought not to be taken too seriously.(a) The
With reference to Figure 3.8, find P(I |D) and P(I |D ), assuming that originally each of the 500 machine parts has the same chance of being chosen for inspection.Data From Figure 3.8 I D 20 10 5

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