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probability and stochastic modeling

Applied Probability And Stochastic Processes 2nd Edition Frank Beichelt - Solutions

Let\[f(x, y)=\frac{1}{2} \sin (x+y), 0 \leq x, y \leq \frac{\pi}{2}\]be the joint probability density of the random vector \((X, Y)\).(1) Determine the marginal densities.(2) Are \(X\) and \(Y\) independent?(3) Determine the conditional mean value \(E(Y \mid X=x)\).(4) Compare the numerical values
The temperatures \(X\) and \(Y\), measured daily at the same time at two different locations, have the joint density\[f_{X, Y}(x, y)=\frac{x y}{3} \exp \left[-\frac{1}{2}\left(x^{2}+\frac{y^{3}}{3}\right)\right], 0 \leq x, y \leq \infty\]Determine the probabilities\[P(X>Y) \text { and } P(X
A large population of rats had been fed with individually varying mixtures of wholegrain wheat and puffed wheat to see whether the composition of the food has any influence on the lifetimes of the rats. Let \(Y\) be the lifetime of a rat and \(X\) the corresponding ratio of wholegrain it had in its
In a forest stand, the stem diameter \(X\) (measured \(1.3 \mathrm{~m}\) above ground) and the corresponding tree height \(Y\) have a bivariate normal distribution with joint density\[f_{X, Y}(x, y)=\frac{1}{0.48 \pi} \exp \left\{-\frac{25}{18}\left(\frac{(x-0.3)^{2}}{\sigma_{x}^{2}}-2 ho
The prices per unit \(X\) and \(Y\) of two related stocks have a bivariate normal distribution with parameters\[\mu_{X}=24, \sigma_{X}^{2}=49, \mu_{Y}=36, \sigma_{Y}^{2}=144, \text { and } ho=0.8\](1) Determine the probabilities\[P(|Y-X| \leq 10) \text { and } P(|Y-X|>15)\]You may make use of
The vector \((X, Y)\) has the joint distribution function \(F_{X, Y}(x, y)\). Show that\[P(X>x, Y>y)=1-F_{Y}(y)-F_{X}(x)+F_{X, Y}(x, y)\]
At time \(t=0\), a parallel system \(S\) consisting of two elements \(e_{1}\) and \(e_{2}\) starts operating. Their lifetimes \(X_{1}\) and \(X_{2}\) are dependent with joint survival function\[\bar{F}\left(x_{1}, x_{2}\right)=P\left(X_{1}>x_{1}, X_{2}>x_{2}\right)=\frac{1}{e^{+0.1
Prove the conditional variance formula\[\operatorname{Var}(X)=E[\operatorname{Var}(X \mid Y)]+\operatorname{Var}[E(X \mid Y)]\]
The random edge length \(X\) of a cube has a uniform distribution in the interval \([4.8,5.2]\). Determine the correlation coefficient \(ho=ho(X, Y)\), where \(Y=X^{3}\) is the volume of the cube.
The edge length \(X\) of a equilateral triangle is uniformly distributed in the interval \([9.9,10.1]\). Determine the correlation coefficient between \(X\) and the area \(Y\) of the triangle.
The random vector \((X, Y)\) has the joint density\[f_{X, Y}(x, y)=8 x y, \quad 0
The random variables \(U\) and \(V\) are uncorrelated and have mean value 0 . Their variances are 4 and 9 , respectively.Determine the correlation coefficient \(ho(X, Y)\) between the random variables\[X=2 U+3 V \text { and } Y=U-2 V\]
The random variable \(Z\) is uniformly distributed in the interval \([0,2 \pi]\). Check whether the random variables \(X=\sin Z\) and \(Y=\cos Z\) are uncorrelated.
A random experiment consists of simultaneously flipping three coins.(1) What is the corresponding sample space?(2) Give the following events in terms of elementary events:\(A=\) 'head appears at least two times,' \(B=\) 'head appears not more than once,' and \(C=\) 'no head appears.'(3)
A random experiment consists of flipping a die to the first appearance of a ' 6 '. What is the corresponding sample space?
Castings are produced weighing either \(1,5,10\), or \(20 \mathrm{~kg}\). Let \(A, B\), and \(C\) be the events that a casting weighs 1 or \(5 \mathrm{~kg}\), exactly \(10 \mathrm{~kg}\), and at least \(10 \mathrm{~kg}\), respectively. Characterize verbally the events \(A \cap B, A \cup B, A \cap
Three randomly chosen persons are to be tested for the presence of gene \(g\). Three random events are introduced:\(A=\) 'none of them has gene \(g\),'\(B=\) 'at least one of them has gene \(g\), '\(C=\) 'not more than one of them has gene \(g\) '.Determine the corresponding sample space \(\Omega\)
Under which conditions are the following relations between events \(A\) and \(B\) true:(1) \(A \cap B=\boldsymbol{\Omega}\), (2) \(A \cup B=\boldsymbol{\Omega}\), (3) \(A \cup B=A \cap B\) ?
Visualize by a Venn diagram whether the following relations between random events \(A, B\), and \(C\) are true:(1) \(A \cap(B \cup C)=(A \cap B) \cup(A \cap C)\),(2) \((A \cap B) \cup(A \cap \bar{B})=A\),(3) \(A \cup B=B \cup(A \cap \bar{B})\).
(1) Verify by a Venn diagram that for three random events \(A, B\), and \(C\) the following relation is true: \((A \backslash B) \cap C=(A \cap C) \backslash(B \cap C)\).(2) Is the relation \((A \cap B) \backslash C=(A \backslash C) \cap(B \backslash C)\) true as well?
The random events \(A\) and \(B\) belong to a \(\sigma-\) algebra \(\boldsymbol{E}\).What other events, generated by \(A\) and \(B\), must belong to \(\boldsymbol{E}\)?
Let \(A\) and \(B\) be two disjoint random events, \(A \subset \boldsymbol{\Omega}, B \subset \boldsymbol{\Omega}\).Check whether the set of events \(\{A, B, A \cap \bar{B}\), and \(\bar{A} \cap B\}\) is (1) an exhaustive and (2) a disjoint set of events (Venn diagram).
A coin is flipped 5 times in a row. What is the probability of the event \(A\) that 'head' appears at least 3 times one after the other?
A die is thrown. Let \(A=\{1,2,3\}\) and \(B=\{3,4,6\}\) be two random events. Determine the probabilities \(P(A \cup B), P(A \cap B)\), and \(P(B \backslash A)\).
A die is thrown 3 times. Determine the probability of the event \(A\) that the resulting sequence of three integers is strictly increasing.
Two dice are thrown simultaneously. Let \(\left(\omega_{1}, \omega_{2}\right)\) be an outcome of this random experiment, \(A={ }^{\prime} \omega_{1}+\omega_{2} \leq 10^{\prime}\) and \(B=' \omega_{1} \cdot \omega_{2} \geq 19\).'Determine the probability \(P(A \cap B)\).
What is the probability \(p_{3}\) to get 3 numbers right with 1 ticket in the ' 6 out of 49 ' number lottery?
A sample of 300 students showed the following results with regard to physical fitness and body weight:One student is randomly chosen. It happens to be Paul.(1) What is the probability that the fitness of Paul is satisfactory?(2) What is the probability that the weight of Paul is greater than \(80
Paul writes four letters and addresses the four accompanying envelopes. After having had a bottle of whisky, he puts the letters randomly into the envelopes. Determine the probabilities \(p_{k}\) that \(k\) letters are in the 'correct' envelopes, \(k=0,1,2,3\).
A straight stick is broken at two randomly chosen positions. What is the probability that the resulting three parts of the stick allow the construction of a triangle?
Two hikers climb to the top of a mountain from different directions. Their arrival time points are between 9:00 and 10:00 a.m., and they stay on the top for 10 and 20 minutes, respectively. For each hiker, every time point between 9 and 10:00 has the same chance to be the arrival time. What is the
A fence consists of horizontal and vertical wooden rods with a distance of \(10 \mathrm{~cm}\) between them (measured from the center of the rods). The rods have a circular sectional view with a diameter of \(2 \mathrm{~cm}\). Thus, the arising squares have an edge length of \(8 \mathrm{~cm}\).
Let \(A\) and \(B\) be disjoint events with \(P(A)=0.3\) and \(P(B)=0.45\). Determine the probabilities \(P(A \cup B), P(\overline{A \cup B}), P(\bar{A} \cup \bar{B})\), and \(P(\bar{A} \cap B)\).
Let \(P(A \cap \bar{B})=0.3\) and \(P(\bar{B})=0.6\). Determine \(P(A \cup B)\).
Is it possible that for two events \(A\) and \(B\) with \(P(A)=0.4\) and \(P(B)=0.2\) the relation \(P(A \cap B)=0.3\) is true?
Check whether for 3 arbitrary random events \(A, B\), and \(C\) the following constellations of probabilities can be true:(1) \(P(A)=0.6, P(A \cap B)=0.2\), and \(P(A \cap \bar{B})=0.5\),(2) \(P(A)=0.6, P(B)=0.4, P(A \cap B)=0\), and \(P(A \cap B \cap C)=0.1\),(3) \(P(A \cup B \cup C)=0.68\) and
Show that for two arbitrary random events \(A\) and \(B\) the following inequalities are true: \(P(A \cap B) \leq P(A) \leq P(A \cup B) \leq P(A)+P(B)\).
Let \(A, B\), and \(C\) be 3 arbitrary random events.(1) Express the event ' \(A\) occurs, but \(B\) and \(C\) do not occur' in terms of suitable relations between these events and their complements.(2) Prove: the probability of the event 'exactly one of the events \(A, B\), or \(C\) occurs'
Two dice are simultaneously thrown. The result is \(\left(\omega_{1}, \omega_{2}\right)\). What is the probability \(p\) of the event ' \(\omega_{2}=6\) ' on condition that ' \(\omega_{1}+\omega_{2}=8\) ?'
Two dice are simultaneously thrown. By means of formula (1.24) determine the probability \(p\) that the dice show the same number.
A publishing house offers a new book as standard or luxury edition and with or without a CD. The publisher analyzes the first 1000 orders:Let \(A(B)\) the random event that a book, randomly choosen from these 1000 , is a luxury one (comes with a CD). (1) Determine the probabilities \(P(A), P(B),
A manufacturer equips its newly developed car of type Treekill optionally with or without a tracking device and with or without speed limitation technology and analyzes the first 1200 orders:Let \(A(B)\) the random event that a car, randomly chosen from these 1200 , has speed limitation (comes with
A bowl contains \(m\) white marbles and \(n\) red marbles. A marble is taken randomly from the bowl and returned to the bowl together with \(r\) marbles of the same color. This procedure continues to infinity.(1) What is the probability that the second marble taken is red?(2) What is the
A test procedure for diagnosing faults in circuits indicates no fault with probability 0.99 if the circuit is faultless. It indicates a fault with probability 0.90 if the circuit is faulty. Let the probability of a circuit to be faulty be 0.02 .(1) What is the probability that a circuit is faulty
Suppose \(2 \%\) of cotton fabric rolls and \(3 \%\) of nylon fabric rolls contain flaws. Of the rolls used by a manufacturer, \(70 \%\) are cotton and \(30 \%\) are nylon.a) What is the probability that a randomly selected roll used by the manufacturer contains flaws?b) Given that a randomly
A group of 8 students arrives at an examination. Of these students 1 is very well prepared, 2 are well prepared, 3 are satisfactorily prepared, and 2 are insufficiently prepared. There is a total of 16 questions. A very well prepared student can answer all of them, a well prepared 12 , a
Symbols 0 and 1 are transmitted independently from each other in proportion \(1: 4\). Random noise may cause transmission failures: If a 0 was sent, then a 1 will arrive at the sink with probability 0.1 . If a 1 was sent, then a 0 will arrive at the sink with probability 0.05 (figure).(1) What is
The companies 1,2 , and 3 have 60,80 , and 100 employees with 45,40 , and 25 women, respectively. In every company, employees have the same chance to be retrenched. It is known that a woman had been retrenched (event \(B\) ).What is the probability that she had worked in company 1,2 , and 3 ,
John needs to take an examination, which is organized as follows: To each question 5 answers are given. But John knows the correct answer only with probability 0.6 . Thus, with probability 0.4 he has to guess the right answer. In this case, John guesses the correct answer with probability \(1 / 5\)
A delivery of 25 parts is subject to a quality control according to the following scheme: A sample of size 5 is drawn (without replacement of drawn parts). If at least one part is faulty, then the delivery is rejected. If all 5 parts are o.k., then they are returned to the lot, and a sample of size
The random events \(A_{1}, A_{2}, \ldots, A_{n}\) are assumed to be independent. Show that\[P\left(A_{1} \cup A_{2} \cup \cdots \cup A_{n}\right)=1-\left(1-P\left(A_{1}\right)\right)\left(1-P\left(A_{2}\right)\right) \cdots\left(1-P\left(A_{n}\right)\right)\]
\(n\) hunters shoot at a target independently of each other, and each of them hits it with probability 0.8 . Determine the smallest \(n\) with property that the target is hit with probability 0.99 by at least one hunter.
Starting a car of type Treekill is successful with probability 0.6 . What is the probability that the driver needs no more than 4 start trials to be able to leave?
Let \(A\) and \(B\) be two subintervals of \([0,1]\). A point \(x\) is randomly chosen from \([0,1]\). Now \(A\) and \(B\) can be interpreted as random events, which occur if \(x \in A\) or \(x \in B\), respectively. Under which condition are \(A\) and \(B\) independent?
A tank is shot at by 3 independently acting anti-tank helicopters with one antitank missile each. Each missile hits the tank with probability 0.6 . If the tank is hit by 1 missile, it is put out of action with probability 0.8 . If the tank is hit by at least 2 missiles, it is put out of action with
An aircraft is targeted by two independently acting ground-to-air missiles. Each missile hits the aircraft with probability 0.6 if these missiles are not being destroyed before. The aircraft will crash with probability 1 if being hit by at least one missile. On the other hand, the aircraft defends
The liquid flow in a pipe can be interrupted by two independent valves \(V_{1}\) and \(V_{2}\), which are connected in series (figure). For interrupting the liquid flow it is sufficient if one valve closes properly. The probability that an interruption is achieved when necessary is 0.98 for both
An ornithologist measured the weight of 132 eggs of helmeted guinea fowls [gram]:There are no eggs weighing less than 38 and more than 50. Let \(X\) be the weight of a randomly picked egg from this sample.(1) Determine the probability distribution of \(X\).(2) Draw the distribution function of
114 nails are classified by length: number i 1 2 3 4 5 6 7 length (in mm) x; number of nails ni < 15.0 15.0 15.1 15.2 15.3 15.4 15.5 15.6 > 15.6 0 3 10 25 40 40 18 16 2 0
A set of 100 coins from an ongoing production process had been sampled and their diameters measured. The measurement procedure allows for a degree of accuracy of \(\pm 0.04 \mathrm{~mm}\). The table shows the measured values \(x_{i}\) and their numbers:Let \(X\) be the diameter of a randomly from
84 specimen copies of soft coal, sampled from the ongoing production in a colliery over a period of 7 days, had been analyzed with regard to ash and water content, respectively [in \%]. Both ash and water content have been partitioned into 6 classes. The table shows the results:Let \(X\) be the
It costs \(\$ 50\) to find out whether a spare part required for repairing a failed device is faulty or not. Installing a faulty spare part causes damage of \(\$ 1000\).Is it on average more profitable to use a spare part without checking if(1) \(1 \%\) of all spare parts of that type,(2) \(3 \%\)
Market analysts predict that a newly developed product in design 1 will bring in a profit of \(\$ 500000\), whereas in design 2 it will bring in a profit of \(\$ 200000\) with probability 0.4 , and a profit of \(\$ 800000\) with probability 0.6 .What design should the producer prefer?
Let \(X\) be the random number one has to throw a die, till for the first time a 6 occurs. Determine \(E(X)\) and \(\operatorname{Var}(X)\).
\(2 \%\) of the citizens of a country are HIV-positive. Test persons are selected at random from the population and checked for their HIV-status.What is the mean number of persons which have to be checked till for the first time an HIV-positive person is found?
Let \(X\) be the difference between the number of head and the number of tail if a coin is flipped 10 times.(1) What is the range of \(X\) ?(2) Determine the probability distribution of \(X\).
A locksmith stands in front of a locked door. He has 9 keys and knows that only one of them fits, but he has otherwise no a priori knowledge. He tries the keys one after the other.What is the mean number of trials till the door opens?
A submarine attacks a warship with 8 torpedoes. The torpedoes hit the warship independently of each other with probability 0.8 . Any successful torpedo hits one of the 8 submerged chambers of the ship independently of other successful ones with probability \(1 / 8\). The chambers are isolated from
Three hunters shoot at 3 partridges. Every hunter, independently of the others, takes aim at a randomly selected partridge and hits his/her target with probability 1 . Thus, a partridge may be hit by several pellets, whereas lucky ones escape a hit.Determine the mean \(E(X)\) of the random number
A lecturer, for having otherwise no merits, claims to be equipped with extrasensory powers. His students have some doubt about it and ask him to predict the outcomes of ten flippings of a fair coin. The lecturer is five times successful. Do you believe that, based on this test, the claim of the
Let \(X\) have a binomial distribution with parameters \(n=5\) and \(p=0.4\).(1) Draw the distribution function of \(X\).(2) Determine the probabilities\[P(X>6), P(X
Let \(X\) have a binomial distribution with parameters \(n=10\) and \(p\).Determine an interval \(\mathbf{I}\) so that \(P(X=2)
The stop sign at an intersection is on average ignored by \(4 \%\) of all cars. A car, which ignores the stop sign, causes an accident with probability 0.01 . Assuming independent behavior of the car drivers:(1) What is the probability that from 100 cars at least 3 ignore the stop sign?(2) What is
Tessa bought a dozen claimed to be fresh-laid farm eggs in a supermarket. There are 2 rotten eggs amongst them. For breakfast she boils 2 eggs.What is the probability that her breakfast is spoilt if already one bad egg will have this effect?
A smart baker mixes 20 stale breads from the previous days with 100 freshly baked ones and offers this mixture for sale. Tessa randomly chooses 3 breads from the 120 , i.e., she does not feel and smell them. What is the probability that she has bought at least one stale bread?
Some of the 270 spruces of a small forest stand are infested with rot (a fungus affecting first the core of the stems). Samples are taken from the stems of 30 randomly selected trees.(1) If 24 trees from the 270 are infested, what is the probability that there are less than 4 infested trees in the
Because it happens that one or more airline passengers do not show up for their reserved seats, an airline would sell 602 tickets for a flight that holds only 600 passengers. The probability that, for some reason or other, a passenger does not show up is 0.008 .What is the probability that every
Flaws are randomly located along the length of a thin copper wire. The number of flaws follows a Poisson distribution with a mean of 0.15 flaws per \(\mathrm{cm}\). What is the probability \(p_{\geq 2}\) of at least 2 flaws in a section of length \(10 \mathrm{~cm}\) ?
The random number of crackle sounds produced per hour by an old radio has a Poisson distribution with parameter \(\lambda=12\).What is the probability that there is no crackle sound during the 4 minutes transmission of a listener's favorite hit?
The random number of tickets car driver Odundo receives has a Poisson distribution with parameter \(\lambda=2\) a year. In the current year, Odundo had received his first ticket on the 31st of March.What is the probability that he will receive another ticket in that year?
Let \(X\) have a Poisson distribution with parameter \(\lambda\).For which nonnegative integer \(n\) is the probability \(p_{n}=P(X=n)\) maximal?
In \(100 \mathrm{~kg}\) of a low-grade molten steel tapping there are on average \(120 \mathrm{impu}-\) rities. Castings weighing \(1 \mathrm{~kg}\) are manufactured from this raw material. What is the probability that there are at least 2 impurities in a casting if the spacial distribution of the
In a piece of fabric of length \(100 \mathrm{~m}\) there are on average 10 flaws. These flaws are assumed to be Poisson distributed over the length. The \(100 \mathrm{~m}\) of fabric are cut in pieces of length \(4 \mathrm{~m}\).What percentage of the \(4 \mathrm{~m}\) cuts can be expected to be
\(X\) have a binomial distribution with parameters \(n\) and \(p\). Compare the following exact probabilities with the corresponding Poisson approximations and give reasons for possible larger deviations:(1) \(P(X=2)\) for \(n=20, p=0.1\),(2) \(P(X=2)\) for \(n=20, p=0.9\),(3) \(P(X=0)\) for
A random variable \(X\) has range \(R=\left\{x_{1}, x_{2}, \cdots, x_{m}\right\}\) and probability distribution\[\left\{p_{k}=P\left(X=x_{k}\right) ; k=1,2, \ldots, m\right\}, \quad \Sigma_{k=1}^{m} p_{k}=1\]A random experiment with outcome \(X\) is repeated \(n\) times. The outcome of the \(k t
A branch of the PROFIT-Bank has found that on average \(68 \%\) of its customers visit the branch for routine money matters (type 1-visitors), \(14 \%\) are there for investment matters (type 2-visitors), 9\% need a credit (type 3 -visitors), \(8 \%\) need foreign exchange service (type 4
Let \(F(x)\) and \(f(x)\) be the respective distribution function and the probability density of a random variable \(X\). Answer with yes or no the following questions:(1) \(F(x)\) and \(f(x)\) can be arbitrary real functions.(2) \(f(x)\) is a nondecreasing function.(3) \(f(x)\) cannot have
Check whether by suitable choice of the parameter \(a\) the following functions are densities of random variables. If the answer is yes, determine the respective distribution functions, mean values, variances, medians, and modes.(1) \(f(x)=a|x|,-3 \leq x \leq+3\),(2) \(f(x)=a x e^{-x^{2}}, x \geq
(1) Show that \(f(x)=\frac{1}{\sqrt{2} x}, 0
Let \(X\) be a continuous random variable. Confirm or deny the following statements:(1) The probability \(P(X=E(X))\) is always positive.(2) There is always \(\operatorname{Var}(X) \leq 1\).(3) \(\operatorname{Var}(X)\) can be negative if \(X\) can assume negative values.(4) \(E(X)\) is never
The current which flows through a thin copper wire is uniformly distributed in the interval \([0,10]\) (in \(m A\) ). For safety reasons, the current should not fall below the crucial level of \(4 \mathrm{~mA}\).What is the probability that at any randomly chosen time point the current is below \(4
According to the timetable, a lecture begins at \(8: 15 \mathrm{a} . \mathrm{m}\). The arrival time of Professor Wisdom in the venue is uniformly distributed between 8:13 and 8:20, whereas the arrival time of student Sluggish is uniformly distributed over the time interval from 8:05 to 8:30.What is
A road traffic light is switched on every day at 5:00 a.m. It always begins with red and holds this colour for two minutes. Then it changes to yellow and holds this colour for 30 seconds before it switches to green to hold this colour for 2.5 minutes. This cycle continues till midnight.(1) A car
A continuous random variable \(X\) has the probability density\[f(x)=\left\{\begin{array}{l} 1 / 4 \text { for } 0 \leq x \leq 2 \\ 1 / 2 \text { for } 2
A continuous random variable \(X\) has the probability density\[f(x)=2 x, 0 \leq x \leq 1\](1) Draw the corresponding distribution function.(2) Determine and compare the measures of variability\[E(|X-E(X)|) \text { and } \sqrt{\operatorname{Var}(X)}\]
The lifetime \(X\) of a bulb has an exponential distribution with a mean value of \(E(X)=8000\) hours. Calculate the probabilities\[P(X \leq 4000), P(X>12000), \quad P(7000 \leq X
The lifetimes of 5 identical bulbs are exponentially distributed with parameter \(\lambda=1.25 \cdot 10^{-4}\left[h^{-1}\right]\).All of them are switched on at time \(t=0\) and will fail independently of each other.(1) What is the probability that at time \(t=8000\) hours a) all 5 bulbs and b) at
The period of employment of staff in a certain company has an exponential distribution with property that \(92 \%\) of staff leave the company after only 16 months. What is the mean time an employee is with this company and the corresponding standard deviation?
The times between the arrivals of taxis at a rank are independent and have an exponential distribution with parameter \(\lambda=4\left[h^{-1}\right]\). An arriving customer does not find an available taxi and the previous one left 3 minutes earlier. No other customers are waiting. What is the
A small branch of a bank has the two tellers 1 and 2 . The service times at these tellers are independent and exponentially distributed with parameter \(\lambda=0.4\left[\mathrm{~min}^{-1}\right]\). When Pumeza arrives, the tellers are occupied by a customer each. So she has to wait. Teller 1 is

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