Chapter 5_Distribution V1_updated.pdf

Chapter 5_Distribution V1_updated.pdf - Nguyen VP Nguyen...

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2/26/2019 1 Department of Industrial & Systems Engineering, HCMUT Email: [email protected] Nguyen VP Nguyen, Ph.D. February 26, 2019 Discrete probability distributions Binomial distribution Multinomial distribution Poisson distribution Hypergeometric distribution Continuous probability distributions Normal distribution Standard normal distribution Gamma distribution Exponential distribution Chi square distribution Lognormal distribution Weibull distribution TAXONOMY OF PROBABILITY DISTRIBUTIONS
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2/26/2019 2 USAGE OF PROBABILITY DISTRIBUTION Distribution (discrete/continuous) function is widely used in simulation studies. A simulation study uses a computer to simulate a real phenomenon or process as closely as possible. The use of simulation studies can often eliminate the need of costly experiments and is also often used to study problems where actual experimentation is impossible. Examples : 1) A study involving testing the effectiveness of a new drug, the number of cured patients among all the patients who use such a drug approximately follows a binomial distribution . 2) Operation of ticketing system in a busy public establishment (e.g., airport), the arrival of passengers can be simulated using Poisson distribution . BINOMIAL DISTRIBUTION In many situations, an outcome has only two outcomes: success and failure . Such outcome is called dichotomous outcome. An experiment when consists of repeated trials, each with dichotomous outcome is called Bernoulli process . Each trial in it is called a Bernoulli trial . Example 4.5: Firing bullets to hit a target. In a Bernoulli process, we define a random variable X ≡ the number of successes in trials. This a random variable obeys the binomial probability distribution, if the experiment satisfies the following conditions: 1)The experiment consists of n trials. 2)Each trial results in one of two mutually exclusive outcomes, one labelled a “success” and the other a “failure” . 3)The probability of a success on a single trial is equal to . The value of ݌ remains constant throughout the experiment. 4)The trials are independent.
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2/26/2019 3 BINOMIAL DISTRIBUTION The function for computing the probability for the binomial probability distribution is given by ݂ ݔ ݊! ݔ! ݊ െ ݔ ! ݌ ሺ1 െ ݌ሻ ௡ି௫ for x = 0, 1, 2, …., n Here, ݂ ݔ ൌ ܲ ܺ ൌ ݔ , where ܺ denotes “the number of success” and ܺ ൌ ݔ denotes the number of success in ݔ trials. Definition: Binomial distribution CRITERIA FOR USING BINOMIAL DISTRIBUTIONS The binomial distribution is used to model the probabilities of occurrences when specific rules are met. Rule #1: There are only two mutually exclusive outcomes for a discrete random variable (i.e., success or failure).
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