PROBABILITY AND STATISTICS WEEK 6 Onur Doğan 2016-2017

Discrete Uniform Distribution A discrete uniform random variable X has an equal probability for each value in the range of X= [a, b], a < b. Thus, the probability mass function of X is; P(x)= 1/(b-a+1) where x=a,a+1,…,b Onur Doğan 2016-2017

Example Casting a die… Onur Doğan 2016-2017

Example Suppose that product codes of 2, 3, or 4 letters are equally likely. Determine the probability mass function of the number of letters (X) in a product code. Calculate the mean and variance of X Onur Doğan 2016-2017

The Binomial Probability Distribution There are many experiments that conform either exactly or approximately to the following list of requirements: 1. The experiment consists of a sequence of n smaller experiments called trials, where n is fixed in advance of the experiment. 2. Each trial can result in one of the same two possible outcomes (dichotomous trials), which we denote by success (S) and failure (F). 3. The trials are independent, so that the outcome on any particular trial does not influence the outcome on any other trial. 4. The probability of success is constant from trial to trial; we denote this probability by p. An experiment for which Conditions 1–4 are satisfied is called a binomial experiment. Onur Doğan 2016-2017

Bernoulli trials A Bernoulli refers to a trial that has only two possible outcomes. (1) Flipping a coin: S = {head, tail) (2) Truth of an answer: S = {right, wrong) (3) Status of a machine: S = {working, broken) (4) Quality of a product: S = {good, defective) (5) Accomplishment of a task: S = {success, failure) A binomial experiment consists of a series of n independent Bernoulli trials with a constant probability of success (p) in each trial. Onur Doğan 2016-2017

Example A seller’s success ? Onur Doğan 2016-2017

The Mean and Variance of X Onur Doğan 2016-2017

Example Suppose that a machine produce defective item with probability 0,1. a) Suppose that machine produces 5 items, find the probability of 1 item defective. b) If the amount of daily production is 100, then what's the expected defective item amount? c)What’s the variance of defective items of samples around the expected defective items. Onur Doğan 2016-2017

Example The probability of making a doctor's successful surgery is %80. If that doctor make 3 surgery in one month, find the all probability for all possible results. Onur Doğan 2016-2017

Example In a certain automobile dealership, 20% of all customers purchase an extended warranty with their new car. For 7 customers selected at random: 1) Find the probability that exactly 2 will purchase an extended warranty 2) Find the probability at most 6 will purchase an extended warranty Onur Doğan 2016-2017

Example Example: Find the mean and standard deviation of the binomial distribution when n = 18 and p = 0.75. Define the probability function. Solutions: 1) n = 18, p = 0.75, q = 1 - 0.75 = 0.25 m = np ( )(0. ) . 18 75 13 5 s » npq 25 3 375 1 8371 P x ( ) (0. = æ è ç ö ø ÷ - 18 75 25 for 0, 1, 2, . , 2) The probability function is: Onur Doğan 2016-2017

The Hypergeometric Distribution The assumptions leading to the hypergeometric distribution are as follows: The population or set to be sampled consists of N individuals, objects, or elements (a finite population). 2. Each individual can be characterized as a success (S) or a failure (F), and there are M successes in the population. 3. A sample of n individuals is selected without replacement in such a way that each subset of size n is equally likely to be chosen. The random variable of interest is X the number of S’s in the sample. The probability distribution of X depends on the parameters n, M, and N, so we wish to obtain P(X x) h(x; n, M, N). Onur Doğan 2016-2017

Example Suppose that a box contains five red balls and ten blue balls. If seven balls are selected at random without replacement, what is the probability that three red balls will be obtained? Onur Doğan 2016-2017

The Hypergeometric Distribution Onur Doğan 2016-2017

Example Suppose that in production line for every 20 products, 4 of them enter reprocessing. a) If we selected 2 products, find the probability of one of them enter reprocessing? b) If we selected 10 products, how many of them should have expected enter reprocessing? Onur Doğan 2016-2017

Note: The hypergeometric distribution is related to the binomial distribution. Whereas the binomial distribution is the approximate probability model for sampling without replacement from a finite dichotomous (S–F) population, the hypergeometric distribution is the exact probability model for the number of S’s in the sample. Onur Doğan 2016-2017

The Negative Binomial Distribution The negative binomial rv and distribution are based on an experiment satisfying the following conditions: 1. The experiment consists of a sequence of independent trials. 2. Each trial can result in either a success (S) or a failure (F). 3. The probability of success is constant from trial to trial, so P(S on trial i)=p for i =1, 2, 3 . . . . 4. The experiment continues (trials are performed) until a total of r successes have been observed, where r is a specified positive integer. Onur Doğan 2016-2017

The Negative Binomial Distribution Onur Doğan 2016-2017

Example A pediatrician wishes to recruit 5 couples, each of whom is expecting their first child, to participate in a new natural childbirth regimen. Let p=P(a randomly selected couple agrees to participate). If p=0.2, what is the probability that 15 couples must be asked before 5 are found who agree to participate? Onur Doğan 2016-2017

The Negative Binomial Distribution Onur Doğan 2016-2017

The Geometric Distributions Onur Doğan 2016-2017

Example In a production line 200 of 1000 items were found to be defective. a)What’s the probability of first defective item is the 4th item tested. b)How many items should have been tested till first defective item found? c)What’s the probability of the first defective item is not the first tested one? Onur Doğan 2016-2017

The Multinomial Distributions Onur Doğan 2016-2017

Example Suppose that there are 3 different brand; A,B and C. And we have probabilities to be purchased; P(A)=0,40 P(B)=0,10 P(C)=0,50 Suppose that there are 10 customers, what’s the probability of 2 of them buy A, 5 of them buy B and 3 of them buy C. Onur Doğan 2016-2017

The Poisson Probability Distribution Onur Doğan 2016-2017

Example Suppose that, in İzmir the number of power blackout has the Poisson distribution with mean 2, for one year. Find the probability of there will be no power blackout in next year? Find the probability of there will be 2 power blackout in next 6 months? Find the probability of there will be 2 or more blackout in next year? Onur Doğan 2016-2017

Example The number of requests for assistance received by a towing service is a Poisson process with rate =4 per hour. a. Compute the probability that exactly ten requests are received during a particular 2-hour period. b. If the operators of the towing service take a 30-min break for lunch, what is the probability that they do not miss any calls for assistance? c. How many calls would you expect during their break? Onur Doğan 2016-2017