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probability and stochastic modeling

Questions and Answers of Probability And Stochastic Modeling

1.5.5 Consider a post office with two clerks. John, Paul, and Naomi enter simultaneously.John and Paul go directly to the clerks, while Naomi must wait until either John or Paul is finished before
1.5.4 A system has two components: A and B. The operating times until failure of the two components are independent and exponentially distributed random variables with parameter 2 for component A,
1.5.3 Let X be an exponentially distributed random variable with parameter . Determine the mean of X(a) by integrating by parts in the definition in equation (1.7) with m D 1;(b) by integrating the
1.5.2 A jar has four chips colored red, green, blue, and yellow. A person draws a chip, observes its color, and returns it. Chips are now drawn repeatedly, without replacement, until the first chip
1.5.1 Let X have a binomial distribution with parameters n D 4 and p D 1 4 . Compute the probabilities PrfX kg for k D 1;2;3; 4, and sum these to verify that the mean of the distribution is 1.
1.4.5 If X follows an exponential distribution with parameter D 2, and independently, Y follows an exponential distribution with parameter D 3, what is the probability that X < Y?
1.4.4 Suppose that the diameters of bearings are independent normally distributed random variables with mean B D 1:005 inch and variance 2 B D .0:003/2 inch2. The diameters of shafts are independent
1.4.3 Let X and Y be independent random variables uniformly distributed over the interval h ???? 12; C 12 ifor some fixed . Show that W D X ????Y has a distribution that is independent of with
1.4.2 Let W be an exponentially distributed random variable with parameter and mean D 1=.(a) Determine PrfW > g.(b) What is the mode of the distribution?
1.4.1 Evaluate the moment EeZ, where is an arbitrary real number and Z is a random variable following a standard normal distribution, by integrating E[eZ] +00 -00 2
1.4.8 Let Z be a random variable with the geometric probability mass function p.k/ D .1????/k; k D 0;1; : : : ;where 0 < < 1.(a) Show that Z has a constant failure rate in the sense that PrfZ D
1.4.7 Given independent exponentially distributed random variables S and T with common parameter , determine the probability density function of the sum R D SCT and identify its type by name.
1.4.6 Suppose that U has a uniform distribution on the interval [0; 1]. Derive the density function for the random variables(a) Y D ????ln.1????U/.(b) Wn D Un for n 1.Hint: Refer to Section 1.2.6.
1.4.5 Let X and Y have the joint normal distribution described in equation (1.47).What value of minimizes the variance of Z D X C.1????/Y? Simplify your result when X and Y are independent.
1.4.4 Twelve independent random variables, each uniformly distributed over the interval.0; 1], are added, and 6 is subtracted from the total. Determine the mean and variance of the resulting random
1.4.3 The lengths, in inches, of cotton fibers used in a certain mill are exponentially distributed random variables with parameter . It is decided to convert all measurements in this mill to the
1.4.2 The median of a random variable X is any value a for which PrfX ag 1 2 and PrfX ag 1 2 . Determine the median of an exponentially distributed random variable with parameter . Compare
1.4.1 The lifetime, in years, of a certain class of light bulbs has an exponential distribution with parameter D 2. What is the probability that a bulb selected at random from this class will last
1.3.16 Consider the generalized geometric distribution defined byandwhere 0 b p=.1????p/.(a) Evaluate p0 in terms of b and p.(b) What does the generalized geometric distribution reduce to when b D
1.3.15 Suppose that X is a Poisson distributed random variable with mean D 2.Determine PrfX g.
1.3.14 Suppose that a random variable Z has the geometric distributionwhere p D 0:10.(a) Evaluate the mean and variance of Z.(b) What is the probability that Z strictly exceeds 10? Pz(k) p(1-p) for
1.3.13 Suppose that a sample of 10 is taken from a day’s output of a machine that produces parts of which 5% are normally defective. If 100% of a day’s production is inspected whenever the sample
1.3.12 Suppose that the telephone calls coming into a certain switchboard during a one-minute time interval follow a Poisson distribution with mean D 4. If the switchboard can handle at most 6
1.3.11 Let X and Y be independent random variables sharing the geometric distribution whose mass function iswhere 0 p(k) = (1-) for k = 0,1,...,
1.3.10 Determine numerical values to three decimal places for PrfX D kg; k D 0;1; 2, when(a) X has a binomial distribution with parameters n D 10 and p D 0:1.(b) X has a binomial distribution with
1.3.9 Suppose that X and Y are independent random variables with the geometric distributionPerform the appropriate convolution to identify the distribution of Z D X CY as a negative binomial. p(k) =
1.3.8 Let X and Y be independent binomial random variables having parameters.N;p/ and .M;p/, respectively. Let Z D X CY.(a) Argue that Z has a binomial distribution with parameters .N CM;p/ by
1.3.7 Let X and Y be independent Poisson distributed random variables having means and , respectively. Evaluate the convolution of their mass functions to determine the probability distribution of
1.3.6 Suppose .X1;X2;X3/ has a multinomial distribution with parameters M andi > 0 for i D 1;2; 3, with 1 C2 C3 D 1.(a) Determine the marginal distribution for X1.(b) Find the distribution for N
1.3.5 Let Y D N ????X where X has a binomial distribution with parameters N and p.Evaluate the product moment E[XY] and the covariance Cov[X;Y].
1.3.4 Let U be a Poisson random variable with mean . Determine the expected value of the random variable V D 1=.1CU/.
1.3.3 Let X be a Poisson random variable with parameter . Determine the probability that X is odd.
1.3.2 The mode of a probability mass function p.k/ is any value k for which p.k/ p.k/ for all k. Determine the mode(s) for(a) The Poisson distribution with parameter > 0.(b) The binomial
1.3.1 Suppose that X has a discrete uniform distribution on the integers 0;1; : : : ; 9, and Y is independent and has the probability distribution PrfY D kg D ak for k D 0;1; : : : . What is the
1.3.6 The discrete uniform distribution on f1; : : : ;ng corresponds to the probability mass function(a) Determine the mean and variance.(b) Suppose X and Y are independent random variables, each
1.3.5 The number of bacteria in a prescribed area of a slide containing a sample of well water has a Poisson distribution with parameter 5. What is the probability that the slide shows 8 or more
1.3.4 A Poisson distributed random variable X has a mean of D 2. What is the probability that X equals 2? What is the probability that X is less than or equal to 2?
1.3.3 A fraction p D 0:05 of the items coming off of a production process are defective.The output of the process is sampled, one by one, in a random manner.What is the probability that the first
1.3.2 A fraction p D 0:05 of the items coming off a production process are defective.If a random sample of 10 items is taken from the output of the process, what is the probability that the sample
1.3.1 Consider tossing a fair coin five times and counting the total number of heads that appear. What is the probability that this total is three?
1.2.13 Let X and Y be independent random variables each with the uniform probability density functionFind the joint probability density function of U and V, where U D maxfX;Yg and V D minfX;Yg. f(x)=
1.2.12 Let U;V, and W be independent random variables with equal variances 2.Define X D U CW and Y D V ????W. Find the covariance between X and Y.
1.2.11 Random variables U and V are independent and have the probability mass functionsDetermine the probability mass function of the sum W D U CV. - Pv (1)= pv(2) 1 2' 1 PU (0) = PU(1) - - PU(2)
1.2.10 Random variables X and Y are independent and have the probability mass functionsDetermine the probability mass function of the sum Z D X CY. Px(0) = 2' Px(3) = PY (1): PY (2)= Py(2) PY(3) =
1.2.9 Determine the mean and variance for the probability mass function p(k) = 2(n-k) n(n - 1) for k 1,2,..., n. ==
1.2.8 Suppose X is a random variable with finite mean and variance 2, and Y D aCbX for certain constants a;b 6D 0. Determine the mean and variance for Y.
1.2.7 Let U and W be jointly distributed random variables. Show that U and W are independent if Pr{U>u and W>w} = Pr{U> u} Pr{W> w} for all u, w.
1.2.6 A pair of dice is tossed. If the two outcomes are equal, the dice are tossed again, and the process repeated. If the dice are unequal, their sum is recorded.Determine the probability mass
1.2.5 Two players, A and B, take turns on a gambling machine until one of them scores a success, the first to do so being the winner. Their probabilities for success on a single play are p for A and
1.2.4 A fair coin is tossed until the first time that the same side appears twice in succession. Let N be the number of tosses required.(a) Determine the probability mass function for N.(b) Let A be
1.2.3 A population having N distinct elements is sampled with replacement. Because of repetitions, a random sample of size r may contain fewer than r distinct elements. Let Sr be the sample size
1.2.2 Let N cards carry the distinct numbers x1; : : : ; xn. If two cards are drawn at random without replacement, show that the correlation coefficient between the numbers appearing on the two
1.2.1 Thirteen cards numbered 1; : : : ;13 are shuffled and dealt one at a time. Say a match occurs on deal k if the kth card revealed is card number k. Let N be the total number of matches that
1.2.10 Let 1A be the indicator random variable associated with an event A, defined to be one if A occurs, and zero otherwise. Define Ac, the complement of event A, to be the event that occurs when A
1.2.9 Determine the distribution function, mean, and variance corresponding to the triangular density. X for 0x1, f(x)=2-x for 1 x2, 0 elsewhere.
1.2.8 A random variable V has the distribution functionwhere A > 0 is a parameter. Determine the density function, mean, and variance. for v < 0, F(v) 1-(1-v)A for 0v1, for v > 1,
1.2.7 Suppose X is a random variable having the probability density functionwhere R > 0 is a fixed parameter.(a) Determine the distribution function FX.x/.(b) Determine the mean E[X].(c) Determine
1.2.6 Let X and Y be independent random variables having distribution functions FX and FY , respectively.(a) Define Z D maxfX;Yg to be the larger of the two. Show that FZ.z/ D FX.z/FY .z/ for all
1.2.5 Let A;B, and C be arbitrary events. Establish the addition law Pr{AUBUC) Pr{A} +Pr{B}+ Pr{C} - Pr{AB) -Pr{AC} Pr{BC} + Pr{ABC}. -
1.2.4 Let Z be a discrete random variable having possible values 0;1; 2, and 3 and probability mass function(a) Plot the corresponding distribution function.(b) Determine the mean E[Z].(c) Evaluate
1.2.3 (a) Plot the distribution function(b) Determine the corresponding density function f .x/ in the three regions (1)x 0, (2) 0 x.(c) What is the mean of the distribution?(d) If X is a random
1.2.2 Let A and B be arbitrary, not necessarily disjoint, events. Establish the general addition lawHint: Apply the result of Exercise 1.2.1 to evaluate PrfABcg D PrfAg????PrfABg. Then, apply the
1.2.1 Let A and B be arbitrary, not necessarily disjoint, events. Use the law of total probability to verify the formulawhere Bc is the complementary event to B (i.e., Bc occurs if and only if B does
6.1. Consider the three server network pictured here:In the long run, what fraction of the time is server #2 idle while, simultaneously, server #3 is busy? Assume that the system satisfies
6.1. In the case m ? 1, n ? 1, verify that 9as given following (6.7)satisfies the equation for the stationary distribution (6.4).
5.1. Suppose three service stations are arranged in tandem so that the departures from one form the arrivals for the next. The arrivals to the first station are a Poisson process of rate A = 10 per
5.2. Refer to the network of Exercise 5.1. Suppose that Server #2 and Server #3 share a common customer waiting area. If it is desired that the total number of customers being served and waiting to
5.1. Consider the three server network pictured here:In the long run, what fraction of time is server #2 idle while, simultaneously, server #3 is busy? Assume that all service times are exponentially
4.5. A ticket office has two agents answering incoming phone calls. In addition, a third caller can be put on HOLD until one of the agents becomes available. If all three phone lines (both agent
4.4. A small grocery store has a single checkout counter with a full-time cashier. Customers arrive at the checkout according to a Poisson process of rate A per hour. When there is only a single
4.3. Balking refers to the refusal of an arriving customer to enter the queue. Reneging refers to the departure of a customer in the queue before obtaining service. Consider an M/M11 system with
4.2. Consider the preemptive priority queue of Section 4.5 and suppose that the arrival rate is k = 4 per hour. Two classes of customers can be identified, having mean service times of 12 minutes and
4.1. Consider the two-server overflow queue of Section 4.4 and suppose the arrival rate is A = 10 per hour. The two servers have rates 6 and 4 per hour. Recommend which server should be placed first.
4.4. Suppose that incoming calls to an office follow a Poisson process of rate A = 6 per hour. If the line is in use at the time of an incoming call, the secretary has a HOLD button that will enable
4.3. Consider a two-server system in which an arriving customer enters the system if and only if a server is free. Suppose that customers arrive according to a Poisson process of rate A = 10
4.2. Customers arrive at a checkout station in a small grocery store according to a Poisson process of rate A = 1 customer per minute. The checkout station can be operated with or without a bagger.
4.1. Consider a two-server system in which an arriving customer enters the system if and only if a server is free. Suppose that customers arrive according to a Poisson process of rate A = 10
3.2. In operating a queueing system with Poisson arrivals at a rate of A = I per unit time and a single server, you have a choice of server mechanisms.Method A has a mean service time of v = 0.5 and
3.1. Let X(t) be the number of customers in an M/GIc queueing system at time t, and let Y(t) be the number of customers who have entered the system and completed service by time t. Determine the
3.5. Let X(t) be the number of customers in an M/G/co queueing system at time t. Suppose that X(0) = 0. Evaluate M(t) = E[X(t)], and show that it increases monotonically to its limiting value as t -
3.4. Customers arrive at a checkout station in a market according to a Poisson process of rate A = 1 customer per minute. The checkout station can be operated with or without a bagger. The checkout
3.3. Customers arrive at a tool crib according to a Poisson process of rate A = 5 per hour. There is a single tool crib employee, and the individual service times are random with a mean service time
3.2. Consider a single-server queueing system having Poisson arrivals at rate A. Suppose that the service times have the gamma densitywhere a > 0 and μ > 0 are fixed parameters. The mean
3.1. Suppose that the service distribution in a single server queue is exponential with rate p.; i.e., G(v) = 1 - e-μY for v > 0. Substitute the mean and variance of this distribution into (3.16)
2.7. Let X(t) be the number of customers in an M/MIcc queueing system at time t. Suppose that X(0) = 0.(a) Derive the forward equations that are appropriate for this process by substituting the birth
2.6. Customers arrive at a service facility according to a Poisson process of rate A. There is a single server, whose service times are exponentially distributed with parameter μ. Suppose that
2.5. Customers arrive at a service facility according to a Poisson process having rate A. There is a single server, whose service times are exponentially distributed with parameter μ. Let N(t) be
2.4. The problem is to model a queueing system having finite capacity.We assume arrivals according to a Poisson process of rate A, independent exponentially distributed service times having mean
2.3. Determine the stationary distribution for an MIM12 system as a function of the traffic intensity p = A/2μ, and verify that L = AW.
2.2. Determine the mean waiting time W for an M/M12 system when A = 2 and μ = 1.2. Compare this with the mean waiting time in an M/M11 system whose arrival rate is A = 1 and service rate is μ =
2.1. Determine explicit expressions for 7r0 and L for the MIMIs queue when s = 2. Plot I - iro and L as a function of the traffic intensity p = A/2μ.
2.3. Customers arrive at a checkout station in a market according to a Poisson process of rate A = 1 customer per minute. The checkout station can be operated with or without a bagger. The checkout
2.2. On a single graph, plot the server utilization 1 - Tro = p and the mean queue length L = p/(1 - p) for the M/M/l queue as a function of the traffic intensity p = A/μ for 0 < p < 1.
2.1. Customers arrive at a tool crib according to a Poisson process of rate A = 5 per hour. There is a single tool crib employee, and the individual service times are exponentially distributed with a
1.1. Two dump trucks cycle between a gravel loader and a gravel unloader.Suppose that the travel times are insignificant relative to the load and unload times, which are exponentially distributed
1.3. Oil tankers arrive at an offloading facility according to a Poisson process whose rate is A = 2 ships per day. Daily records show that there is an average of 3 ships unloading or waiting to
1.2. Consider a system, such as a barber shop, where the service required is essentially identical for each customer. Then actual service times would tend to cluster near the mean service time. Argue
1.1. What design questions might be answered by modeling the following queueing systems?The Customer The Server(a) Arriving airplanes The runway(b) Cars A parking lot(c) Broken TVs Repairman(d)
5.4. In the Ehrenfest urn model (see III, Section 3.2) for molecular diffusion through a membrane, if there are i particles in urn A, the probability that there will be i + 1 after one time unit is 1
5.3. Verify the option valuation formulation (5.10).Hint: Use the result of Exercise 4.6.

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