3) Extra Credit Challenge: We have seen in class that the Poisson distribution with parameter A = np isa good approximation to the probability distribution of the number of successes in n independent trials when eachtrial has probability p of being a success, provided that n is large and p small. In fact, the Poisson distributionremains a good approximation even when the trials are not independent, provided that their dependence is weak.Furthermore, it is not required that each trial be identical. For example, if the i-th trial had a probability p; ofbeing a success (i = 1,2, ...,n), then the total number of successes that occur in the n trials is well approximatedby a Poisson random variable with A = n E, Pi, if each Pi is "small" and each trial is independent or only"weakly dependent." This is called the Poisson paradigm, .Suppose an experiment has n identical and independent trials, with each trial having k possible outcomes withprobabilities p1,...,Pk respectively. Using the Poisson paradigm, show that if all the p,'s are small then theprobability that no trial outcome occurs more than once is approximately equal to exp(-n(n – 1)E P/2).

Poisson distribution is a limiting case of Binomial distribution under the following conditions ;…

Answered: 3) Extra Credit Challenge: We have seen…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Suppose that we have a sample space S = {E₁, E2, E3, E4, E5, E6, E7], where E₁, E2, ..., E7 denote the sample points. The following probability assignments apply: P(E₁) = 0.20, P(E₂) = 0.05, P(E3) = 0.20, P(E4) = 0.15, P(E5) = 0.25, P(E6) = 0.05, and P(E7) = 0.10. Let A = {E₁, E4, E6} B = {E2, E4, E7} C = {E2, E3, E5, E7}. Find P(AC). 0.7000 0.6000 0.3000 0.5714 O 0.4000 Question 3
.11 Suppose X is a continuous-type random variable with CDF Fx. Let Y be the result of applying Fx to X. that is, Y = Fx(X). Find the distribution of Y.
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Aa.127. Let X1,X2,...,X4 be indepedent Poisson random variables with rates λ1,...,λ4, respectively. Show that X1 + X2 + X3 + X4 is a Poisson random variable with rate λ1 + λ2 + λ3 + λ4.
A statistics professor finds that when he schedules an office hour at the 10:30 a.m. time slot, an average of three students arrives. Use the Poisson distribution to find the probability that in a randomly selected office hour no students will arrive. O A. 0.1225 O B. 0.0743 O C. 0.1108 O D. 0.0498 O Time Remaining: 01:08:58 Next TCJT 11 Test 2 1-45 of 45
9. This is a simplified inventory problem. Suppose that it costs c dollars to stock anitem and that the item sells for s dollars. Suppose that the number of items thatwill be asked for by customers is a random variable with the frequency functionp(k). Find a rule for the number of items that should be stocked in order tomaximize the expected income. (Hint: Consider the difference of successiveterms.)
A barbershop is operated by one barber and has a space capacity to accommodate for a total of 6 customers (including the one in service). If a customer comes to this shop and finds it full, he goes to other shops. Customers arrive according to Poisson distribution at a rate of 4.1 customers per hour. The barber’s service time is exponential and he can serve at a rate of 7.1 customers per hour. a. What is the probability that the barber is free? Answer = Answer b. What is the expected number of people in the system? Answer = Answer c. What is the expected waiting time in the queue? Answer = Answerhours. (hint: use effective arrival rate)
The data records the smiling times, in seconds, of an eight-week-old baby.We will assume that the smiling times, in seconds, follow a uniform distribution between zero and 23 seconds, inclusive. This means that any smiling time from zero to and including 23 seconds is equally likely.What is the probability that a randomly chosen eight- week-old baby smiles between two and 18 seconds? O 0.1304 O 0.8696 O 0.6957 0.7826

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