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MTHE 224

Week 3: Probability Distributions Goals: • Study binomial distributions • Study probabilities of continuous random variables Suggested Textbook Readings: Chapter 3: 3.4-3.5, Chapter 4: 4.1-4.2 Practice Problems: Section 3.4: 49, 50, 51, 65 Section 3.5: 75 Section 4.1: 5, 8 Section 4.2: 11, 13, 15, 19 Week 3: Probability Distributions 2 The Binomial Distribution Many experiments result in outcomes that can be classified in one of the two categories, success or failure. Such experiments are called binomial experiments. An binomial experiment has to satisfy the following conditions. 1. a sequence of n trials (small experiments) are repeated, 2. each trial has only two possible outcomes, success (S) and failure (F), 3. all trials are independent, 4. probability of events does not change from trial to trial. The binomial random variable X associated with a binomial experiment consisting of n trials is defined as X = the number of S’s among the trials. The binomial distribution need two parameters: • n, the number of trials • p, the probability of one of the outcomes (usually the success) Knowing n and p, we can compute directly the probability of x successful trials (pmf), denoted by b(x;n,p),  ▯ ▯  n p (1 − p)n−x,x =0 ,1,2,··· ,n b(x;n,p)=  x 0, otherwise Binomial distribution is often denoted by X ∼ bin(n,p). Example 1: (Example 3.32, page 112) Suppose that 20% of all copies of a textbook fail a certain binding strength test. Now we randomly choose 15 copies, and let X be the number of copies that fail the test. (a) What is the probability that exact 6 copies fail the test? MTHE 224 Fall 2012 Week 3: Probability Distributions 3 We may use following commands to find pmf of a binomial distribution in MatLab. n=15; p=0.2; x=0:n; Px=binopdf(x,n,p); bar(x,Px) [Example 1: continued] (b) What is the probability that at most 6 copies fail the test? CDF of a Binomial Distribution For a binomial random variaxle X, the cdf will be denoted by P(X ≤ x)= B(x;n,p)= ▯ b(y;n,p),x =0 ,1,2,··· ,n y=0 MatLab Command: binocdf(x,n,p) (c) What is the probability that at least 6 copies (6 or more) fail the test? MTHE 224 Fall 2012 Week 3: Probability Distributions 4 [Example 1: continued] (d) What is the expected number of copies that fail the test? MatLab Simulations p=0.2; n=15; E=0; for x=0:n E=E+ x*binopdf(x,n,p); end display(E); E= 3.0000 (e)What is the variance and standard deviation of this binomial distribution? Expectation and Variance of Binomial Distributions E(X)= np, V (X)= np(1 − p) MTHE 224 Fall 2012 Week 3: Probability Distributions 5 The Negative Binomial Distribution The negative binomial random variable and distribution are based on an experiment satisfying the following conditions: 1. the experiment consists of a sequence of independent trials, 2. each trial has two possible outcomes, success or failure, 3. probability of success is constant, 4. the trials continue until a total of r successes have been observed. In contrast to the binomial rv, the number of successes is fixed and the number of trials is random. Example 2: An oil company conducts a geological study that indicates that an ex- ploratory oil well should have a 20% chance of striking oil. (a) What is the probability that the first strike comes on the fifth well drilled? (b) What is the probability that the fourth strike comes on the tenth well drilled? The pmf of the negative binomia
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