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BBA182 Applied Statistics Week 7 (2) Discrete Probability Distributions: Binomial and Poisson Distribution Dr Susanne Hansen Saral Email: susanne.saral@okan.edu.tr

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Mid-term exam 23/03/2017 11:45 – 13:00 hours Bring: Calculator Pen Eraser

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Probability and cumulative probability distribution of a discrete random variable In the last class we saw how to calculate the probability of a specific discrete random variable, such as P( x = 3) and the cumulative probability such as P(x 3) in n trials with the following formula: We need to know n, x and P (probability of success)

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The binomial distribution is used to find the probability of a specific or cumulative number of successes in n trials.

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Probability and cumulative probability distribution of a discrete random variable Ali, real estate agent has 5 potential customers to buy a house or apartment with a probability of 0.40. The probability and cumulative probability table of the sale of houses here below. We are able to answer what is P( x=3), P(x 4), P(x >3)etc.

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Shape of Binomial Distribution The shape of the binomial distribution depends on the values of P and n

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Binomial Distribution shapes When P = .5 the shape of the distribution is perfectly symmetrical and resembles a bell-shaped (normal distribution) When P = .2 the distribution is skewed right. This skewness increases as P becomes smaller. When P = .8, the distribution is skewed left. As P comes closer to 1, the amount of skewness increases.

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Using Binomial Tables instead of to calculating Binomial probabilities manually

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Solving Problems with Binomial Tables MSA Electronics is experimenting with the manufacture of a new USB-stick and is looking into the number of defective USB-sticks Every hour a random sample of 5 USB-sticks is taken The probability of one USB-stick being defective is 0.15 What is the probability of finding 3, 4, or 5 defective USB-sticks ? P( x = 3), P(x = 4 ), P(x= 5)

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Solving Problems with Binomial Tables

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Solving Problems with Binomial Tables MSA Electronics is experimenting with the manufacture of a new USB-stick Every hour a random sample of 5 USB-sticks is taken The probability of one USB-stick being defective is 0.15 What is the probability of finding more than 3 (3 inclusive) defective USB-sticks? P( x 3)

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Solving Problems with Binomial Tables Cumulative probability

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Solving Problems with Binomial Tables Cumulative probabilities

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Suppose that Ali, a real estate agent, has 10 people interested in buying a house. He believes that for each of the 10 people the probability of selling a house is 0.40. What is the probability that he will sell 4 houses, P(x = 4)?

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Suppose that Ali, a real estate agent, has 10 people interested in buying a house. He believes that for each of the 10 people the probability of selling a house is 0.20. What is the probability that he will sell 7 houses, P(x = 7)?

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Suppose that Ali, a real estate agent, has 10 people interested in buying a house. He believes that for each of the 10 people the probability of selling a house is 0.35. What is the probability that he will sell more than 7 houses, 7 houses included, P(x 7)?

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Poisson random variable, first proposed by Frenchman Simeon Poisson (1781-1840)

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Poisson Random Variable - three requirements 1. The number of expected outcomes in one interval of time or unit space is unaffected (independent) by the number of expected outcomes in any other non-overlapping time interval. Example: What took place between 3:00 and 3:20 p.m. is not affected by what took place between 9:00 and 9:20 a.m.

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Poisson Random Variable - three requirements (continued) 2.The expected (or mean) number of outcomes over any time period or unit space is proportional to the size of this time interval. Example: We expect half as many outcomes between 3:00 and 3:30 P.M. as between 3:00 and 4:00 P.M. 3.This requirement also implies that the probability of an occurrence must be constant over any intervals of the same length. Example: The expected outcome between 3:00 and 3:30P.M. is equal to the expected occurrence between 4:00 and 4:30 P.M..

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Examples of a Poisson Random variable The number of cars arriving at a toll booth in 1 hour (the time interval is 1 hour) The number of failures in a large computer system during a given day (the given day is the interval) The number of delivery trucks to arrive at a central warehouse in an hour. The number of customers to arrive for flights at an airport during each 10-minute time interval from 3:00 p.m. to 6:00 p.m. on weekdays

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Situations where the Poisson distribution is widely used: Capacity planning – time interval Areas of capacity planning observed in a sample: A bank wants to know how many customers arrive at the bank in a given time period during the day, so that they can anticipate the waiting lines and plan for the number of employees to hire. At peak hours they might want to open more guichets (employ more personnel) to reduce waiting lines and during slower hours, have a few guichets open (need for less personnel).

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Poisson Probability Distribution Poisson Probability Function

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Example: Drive-up ATM Window Poisson Probability Function: Time Interval Suppose that we are interested in the number of arrivals at the drive-up ATM window of a bank during a 15-minute period on weekday mornings. If we assume that the probability of a car arriving is the same for any two time periods of equal length and that the arrival or non-arrival of a car in any time period is independent of the arrival or non-arrival in any other time period, the Poisson probability function is applicable. Then if we assume that an analysis of historical data shows that the average number of cars arriving during a 15-minute interval of time is 10, the Poisson probability function with  = 10 applies.

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Using Poisson ProbabilityTables

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The shape of a Poisson Probabilities Distribution

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Poisson Distribution Shape The shape of the Poisson Distribution depends on the parameter  :

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Continuous Probability Distributions Uniform Probability Distribution Normal Probability Distribution Exponential Probability Distribution

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Continuous random variable A continuous random variable can assume any value in an interval on the real line or in a collection of intervals. It is not possible to talk about the probability of the random variable assuming a particular value, because the probability will be close to 0. Instead, we talk about the probability of the random variable assuming a value within a given interval.

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Probability Density Function The probability density function, f(x), of a continuous random variable X has the following properties: f(x) > 0 for all values of x The area under the probability density function f(x) over all values of the random variable X within its range, is equal to 1.0 The probability that X lies between two values is the area under the density function graph between the two values

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Probability Density Function The probability density function, f(x), of random variable X has the following properties: The cumulative density function F(x0) is the area under the probability density function f(x) from the minimum x value up to x0 where xm is the minimum value of the random variable x

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Probability as an Area

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Probability as an Area

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Cumulative Distribution Function, F(x) Let a and b be two possible values of X, with a < b. The probability that X lies between a and b is:

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Cumulative probability as an Area

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Probability Distribution of a Continuous Random Variable

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The Normal Distribution The Normal Distribution is one of the most popular and useful continuous probability distributions The probability density function:

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‘Bell Shaped’ ‘Bell Shaped’ Symmetrical Mean, Median and Mode are Equal Location of the curve is determined by the mean, μ Spread is determined by the standard deviation, σ The random variable has an infinite theoretical range: +  to  

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The Normal Distribution Symmetrical with the midpoint representing the mean Shifting the mean does not change the shape Values on the X axis are measured in the number of standard deviations away from the mean 1 2 3 As standard deviation becomes larger, curve flattens As standard deviation becomes smaller, curve becomes steeper

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Probability as Area Under the Curve

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Finding Normal Probabilities

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Finding Normal Probabilities

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The Standard Normal Distribution – z-values Any normal distribution (with any mean and standard deviation combination) can be transformed into the standardized normal distribution (Z), with mean 0 and standard deviation 1 We say that Z follows the standard normal distribution.

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Using the Standard Normal Table Step 1 Convert the normal distribution into a standard normal distribution Mean of 0 and a standard deviation of 1 The new standard random variable is Z:

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Using the Standard Normal Table For m = 100, s = 15, find the probability that X is less than 130 = P(x < 130) Transforming x - random variable into a z - standard random variable:

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Using the Standard Normal Table Step 2 Look up the probability from the table of normal curve areas Column on the left is Z value Row at the top has second decimal places for Z values

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Using the Standard Normal Table

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Haynes Construction Company Builds three- and four-unit apartment buildings: Total construction time follows a normal distribution For triplexes, m = 100 days and = 20 days Contract calls for completion in 125 days Late completion will incur a severe penalty fee Probability of completing in 125 days? P(x