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Random variables. Distributions. ch2. Random variables and. their distributions. 2.3. Some Important Discrete. Probability Distributions. Discrete Probability Distributions. Uniform. We have a finite set of outcomes. each of. which has the same probability of occurring (equally.

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**Random variables**Distributions ch2 Random variables and their distributions**2.3**Some Important Discrete Probability Distributions Discrete Probability Distributions**Uniform**We have a finite set of outcomes each of which has the same probability of occurring (equally likely outcomes). So X is said to have a Uniform distribution and we Write**Solution:**Example 2.2 When a light bulb is selected at random from a box that contains a 40-watt bulb, a 60-watt bulb, a 75-watt bulb, and a 100-watt bulb. Find Example 2.3 When a die is tossed, S={1,2,3,4,5,6}. P(each element of the sample space) = 1/6. Therefore, we have a uniform distribution, with**Bernoulli trial**伯努利资料 A Bernoulli trial is an experiment which has two possible outcomes: ‘success’ and ‘failure’. Let p = P(success), q = P (failure ) (q=1-p). The pmf of X is or**so that**Binomial Consider a sequence of n independent Bernoulli trials each of which must result in either a ‘success’ with probability of p or a ‘failure’ with probability q=1-p. Let X= the total number of successes in these n trial P( the total number of x successes) = X is said to have a Binomial distribution with parameters n and p and we write X~Bin(n, p) or X~b(x;n, p)**B(n,p)**b(1, p) X is said to have a Binomial distribution with parameters n and p and we write X~Bin(n, p) or X~b(x;n, p) Special case, when n=1,we have We write**Solution:**Solution: Example 2.4 The probability that a certain kind of component will survive a given shock test is 3/4. Find the probability that exactly 2 of the next 4 components tested survive. Example 2.5 The probability that a patient recovers from a rare blood disease is 0.4. If 15 people are known to have contracted this disease, what is the probability that (a) at least 10 (b) from 3 to 8 survive, survive, and (c) exactly 5 survive?**Example 2.6**A large chain retailer purchases a certain kind of electronic device from a manufacturer. The manufacturer indicates that the defective rate of the device is 3%. Solution： (a) The inspector of the retailer randomly picks 20 items from a shipment. What is the probability that there will be at least one defective item among these 20? (b) Suppose that the retailer receives 10 shipments in a month and the inspector randomly tests 20 devices per shipment. What is the probability that there will be 3 shipments containing at least one defective device?**distribution with parameter is given by**Poisson The pmf of a random variable X which has a Poisson and we write 泊松资料**特大洪水**地震 火山爆发 商场接待的顾客数 电话呼唤次数 交通事故次数**二项分布泊松分布**单击图形播放/暂停 ESC键退出**Solution：**Example 2.7 During a laboratory experiment the average number of radioactive particles passing through a counter in 1 millisecond is 4. What is the probability that 6 articles enter the counter in a given millisecond?**Solution：**Example 2.8 Ten is the average number of oil tankers arriving each day a certain port city. The facilities at the port can handle at most 15 tankers per day. What is the probability that on a given day tankers have to be turned away?**Solution:**each element of the sample space occurs with probability 1/4. Therefore, we have a uniform distribution:**Solution:**Assuming that the tests are independent and p=3/4 for each of the 4 tests, we obtain**Solution：**Let X = the number of people that survive. (a) (b)**28**Solution： (a) Denote by X the number of defective Devices among the 20. Hence Then this X follows a b (x; 20,0.03). (b) Assuming the independence from shipment to shipment and denoting by Y . Y=the number of shipments containing at least one defective. Then Y~b(y;10,0.4562). Therefore,**Solution：**Using the Poisson distribution with x=6 and , We find from Table 1 that**Solution：**Let X be the number of tankers arriving each day. using Table, we have Then,**伯努利资料**Jacob Bernoulli Born: 27 Dec 1654 in Basel, SwitzerlandDied: 16 Aug 1705 in Basel, Switzerland**泊松资料**Siméon Poisson Born: 21 June 1781 in Pithiviers, FranceDied: 25 April 1840 in Sceaux (near Paris), France

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