Problem 1.73 A laboratory keeps a population of aphids. The probability of an aphidpassing a day uneventfully is q< 1. Given that a day is not uneventful, there is a probabilityr that the aphid will have one offspring, a probability s that it will have two offspring,and a probability t that it will die of exhaustion, where r+s+t=1. Offspring are readyto reproduce the next day. The fates of different aphids are independent, as are the eventsof different days. The laboratory starts out with one aphid.Let X, be the number of aphids at the end of n days. Show how to obtain an expressionfor the PGF d(z) of X„. What is the expected value of X,?Show that the probability of extinction does not depend on q and that if 2r + 3s <1,the aphids will certainly die out. Find the probability of extinction if r=1/5, s=2/5 and1= 2/5.

The Provided information is ∅xn(z) is the PGF of xn The number of aphids at the end of the day.…

Answered: Problem 1.73 A laboratory keeps a…

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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Question 7: Let X be a random variable such that E(X – 2)² = 10 and E(X+1)² = 4, then а. Е(X) b. Е(X) %3D с. Е (X) %3D d. E(X) =
Example 2.4.6 A series of tests on a certain type of electrical relay, have revealed that in approximately 5% of the trials, the relay fails to operate under the specified conditions. What is the probability that in ten trials made under these conditions the relay will fail to operate one or more times?
Rework problem 21 from section 4.1 of your text involving the deep-sea fisherman. Assume that the fisherman can use between 5 and 8 lines. With 5 lines, the probability of a catch on each line is 0.77. With 6 this probability is 0.66, with 7 it is 0.52, and with 8 the probability is 0.49. How many lines should the fisherman use to maximize the probability of catching at least 3 fish?
Answer question 8.1
Problem 1.23 You play a match against an opponent in which at each point either you or he/she serves. If you serve you win the point with probability p,, but if your opponent serves you win the point with probability p2. There are two possible conventions for serving: (i) alternate serves; (ii) the player serving continues until he/she loses a point. You serve first and the first player to reach n points wins the match. Show that your probability of winning the match does not depend on the serving convention adopted.
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Rework problem 29 from section 4.1 of your text about the professor who sometimes forgets to bring her briefcase to the office, but assume that, each day, the probability that she forgets the briefcase is 1/8. Assume that her forgetting is a Bernoulli process. (1) What is the probability that she remembers to bring her briefcase every day in one week (5 days)? (2) What is the probability that she forgets to bring her briefcase every day in one week (5 days)? (3) What is the probability that she forgets to bring her briefcase at least one day in one week (5 days)?
Problem 2 [12 points]: Fourth digit of ID a. Genuine motor company manufactures an assembly which consists of three mechanical components. Suppose that the probability of not meeting specification for the first component is equal to 0.09/ 0.11/ 0.13. For the second and third components the probabilities of not meeting specifications are 0.05/ 0.04/ 0.03 and 0.06/ 0.07/ 0.08 respectively. Assuming that the components are independent in terms of not meeting the specification, determine the probability mass function of the number of components in the assembly that does not meet specification. (6.0) Fifth digit of ID Last digit of ID
QUESTION 2 Three local textile distributors (K, L, and M) are competing for a contract to supply textiles to Company XYZ. The probabilities that textile distributors K, L, and M will win the contract are 0.4, 0.3, and 0.3, respectively. If textile distributors K, L, and M win the contract, the probabilities that they will make profits are 0.65, 0.85, and 0.45, respectively. a) Draw a tree diagram for the above information. b) Calculate the probability that the contract was awarded to textile distributor K given that the contract is found to be unprofitable.
a. A researcher studied the effect of social influence on drinking behaviour. Sixty people were randomly assigned to one of three conditions where they were given a soft drink to consume while waiting for the experiment to supposedly begin. An experimental assistant, posing as another participant, was also given a drink. In one condition, the assistant sipped the drink at a faster rate than the research participant; in another condition, the assistant sipped the drink at the same rate as the participant; and in the last condition, the assistant sipped the drink at a slower rate than the participant. The time it took each participant to consume his or her drink was measured.
Question 4 Suppose that in a population of twins, males (M) and females (F) are equally likely to occur and that the probability that twins are identical is 0. If twins are not identical, their genders are independent (if they are identical, their genders are the same). (i) Show that, ignoring birth order, P(MM) = P(FF) = (1+0)/4 and P(MF) = (1–0)/2. (ii) Suppose that n twins are sampled. It is found that n₁ are MM, n2 are FF, and në are MF, but it is not known which twins are identical. Find the maximum likelihood estimator of 0.
Example 7.40: You set out on a walk with three possible locations, denoted locations 1, 2 and 3. The probabilities of travelling from a specified location to the next are given by Solution. If you know that you started in location 3, what is the probability that you end up at location 1 after two time periods? Using the above equation, we get To solve this we use the equation Xn+1 = AXn. Given that you began in location 3, the initial state vector is and subsequently Exercise X₁ = AX0 Probability = A = = X₂ = AX₁ 0.5 0.2 0.4 0.4 0.6 0.1 0.1 0.2 0.5 = Xo 0 0.5 0.2 0.4 0.4 0.6 0.1 0.2 0.5 0.1 ][ Therefore the probability of ending up in location 1 after two time periods is 0.42. 0.5 0.2 0.4 0.4 0.6 0.1 0.1 0.2 0.5 1 0.4 0.1 = 0.5 0.4 0.1 0.5 = 0.42 0.27 0.31 Refer to the example above. If you know that you started in location 2, what is the probability that you will end up at location 1 after one time period?

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