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The mean number of customers appearing in a bank during a self-assertively picked hour is four. The bank boss is contemplating reducing the amount of tellers. however, she should be sure that lines don't get unreasonably long. She infers that if near two customers come in during a 15-minute stretch of time. two tellers (instead of the current three) will be sufficient. a What is the probability near two customers will come in the bank during an indiscriminately picked 15-minute time span? b What is the probability that different customers will come in during two consecutive 15-minute intervals of time? c The director records the amount of customers coming in during 15-minute time frames until she sees a time interval during which various customers appear. Eight time periods have been recorded, each with two or less customers appearing in each. What is the probability that more than 14 time periods will be seen before having different customers appearing during an organized 15-minute term.

Question 66

A particular sort of microorganisms cell detaches at a reliable rate An as time goes on. Thusly, the probability that a particular cell will segregate in a little interval of time t is generally At. Given that a general population starts at time zero with k cells of this sort, and cell divisions are self-ruling of one another, the size of the general population at time t, X (t), has the probability spread

(t) = it] = (11 - I ) e-Akt k - 1

1 _ e-ity-k

71 =

a Find the ordinary worth of X (t) to the extent An and t. b If, for a particular kind of microorganisms cell, A = 0.1 each second, and the general population starts with two cells at time zero, find the typical people size following 5 seconds.

Question 67

During World War I, the British government set up the Industrial Fatigue Research Board (IFRB), later known as the Industrial Health Research Board (IHRB) (Haight 2001). The board was made because of stress for the tremendous number of accidental passings and wounds in the British clash creation undertakings. One of the enlightening lists they considered was the amount of setbacks experienced by women managing 6-inch shells during the period February 13, 1918 to March 20, 1918. These are appeared in the table that follows. Subsequently, 447 women had no accidents during this time span. regardless, 2 had in any occasion 5 setbacks. Number of Frequency Accidents Observed

0 447 1 132 2 42 3 4 3 5 or more a Find the ordinary number of accidents a woman had during this time period. (Anticipate that all discernments in the characterization - at any rate 5" are really 5.)

b After created by von Bortkiewicz (see Section 4.7.3), the Poisson movement had been applied to innumerable self-assertive miracles and, with few uncommon cases, had been found to portray the data well. This had driven the Poisson allocation to be known as the 'unpredictable dissemination," a term that is at this point found in the composition. Along these lines, the mathematicians at the IFRB began by showing these data using the Poisson movement. Find the typical number of women having 0, 1, 2, 3, 4, and 5 incidents using the mean found somewhat (a) and the Poisson scattering. How well do you think this model portrays the data?

c Greenwood and Woods (1919) prescribed fitting a negative binomial assignment to these data. Find the typical number of approved drivers having 0,1, 2, 3, 4, and 5 setbacks using the mean found in part (a) and the numerical (negative binomial with r= 1) dispersal. How well do you think this model depicts the data? Credible note: Researchers were puzzled with respect to why the negative binomial fit better contrasted with the Poisson appointment until, in 1920, Greenwood and Yule proposed the going with model. Expect that the probability any given approved driver will have a setback is passed on by a Poisson scattering with mean A. Nevertheless, A vacillates starting with one woman then onto the next as demonstrated by a gamma transport (see Chapter 5). By then the amount of disasters would have a negative binomial apportionment. The value of A related with an approved driver was called their - disaster tendency."

Question 68

In the game Lotto 6-49, six numbers are discretionarily picked without replacement from 1 to 49. A player who organizes with all of the six numbers in any solicitation wins the large stake.

a What is the probability of winning any given gold mine with one game ticket? b If a game ticket costs $1.00, what are the for the most part expected prizes from playing Lotto 6-49 once. c Suppose an individual gets one Lotto 6-49 ticket each week for seemingly forever. Expecting all years have 52 weeks, what is the probability of prevailing in any event one treasure trove during this time? (Hint: Use a Poisson surmise.) d Given the setting halfway (c), what are the for the most part expected rewards over 100 years?

Question 69

The probability of a customer's appearance up at a fundamental food thing organization counter in any one second reciprocals 0.1. Expect that customers appear in a discretionary stream and, along these lines, that the arrival in any one second is self-sufficient of some other appearance. In like manner acknowledge that at most one customer can appear during any one second.

a Find the probability that the essential appearance will occur during the third 1-second range. b Find the probability that the fundamental appearance will not occur until in any occasion the third 1-second stretch. c Find the probability that no appearances will occur in the underlying 5 seconds. d Find the probability that in any occasion three people will appear in the underlying 5 seconds.

5. [12 marks] Consider the Markov chain with the following transition matrix. 0 0.5 0.5 0.5 0 0.5 0.5 0.5 0 (a) Draw the transition diagram of the Markov chain. (b) Is the Markov chain ergodic? Give a reason for your answer, (c) Compute the two step transition matrix of the Markov chain. (d) What is the state distribution 72 for t = 2 if the initial state distribution for t = 0 is no = (0.1, 0.5, 0.4) ?? (e) What is the average time /1,1 for the chain to return to state 1? Remember: Mij = 1 +) Pikikj .Mr. Adam is a coffee addict. He keeps switching between three brands of coffee, say A, B and C, from week to week according to a discrete Markov chain with the following transition probability matrix: 0.2 0.3 0.57 P = 0.1 0 0.9 0.55 0 0.45 1. Check whether the Markov chain is an irreducible chain or not? 2. Is the Markov chain is periodic or aperiodic? 3. Is the Markov chain has a unique stationary distribution? 4. If Mr. Adam is using brand A this week (ie., week 1), what is the probability distribution of the brand he will be using in week 20?2. (10 points) Consider a continuous-time Markov chain with the transition rate matrix 4 2 2 Q = 3 1 CT 0 (a) What is the expected amount of time spent in each state? (b) What is the transition probability matrix of the embedded discrete-time Markov chain? (c) Is this continuous-time Markov chain irreducible? (d) Compute the stationary distribution for the continuous-time Markov chain and the em- bedded discrete-time Markov chain and compare the two