Chapter 05-Lecture Notes PDF

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Professor Khatereh Ahadi...

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OPRE 3360: Chapter 5 - Anderson Discrete Probability Distribution

In Chapter 4, we defined the concept of an experiment and its associated experimental outcomes. A random variable provides a means for describing experimental outcomes using numerical values. Random variables must assume numerical values.

Random variable: A random variable is a function or rule that assigns a number to each outcome of an experiment. Alternatively, the value of a random variable is a numerical event. Instead of talking about the coin flipping event as {heads, tails} think of it as “the number of heads when flipping a coin”: {0, 1} Discrete random variable: A random variable which can only take a countable number of values. Example: Let X be the number of heads observed in an experiment that flips a coin 10 times, then the values of X are 0, 1, 2, · · · , 10 Example: Let X be the number of customers arriving to a store in a day, where X can take on the values 0, 1, 2, · · · . Continuous random variable: A random variable which can take infinitely many values. Example: Let X as the time to write a statistics exam where the time limit is 3 hours and students cannot leave before 30 minutes.

Note: Integers are Discrete, while Real Numbers are Continuous.

Example: Identify the following random variables as discrete or continuous. Daily return on a stock Number of customers waiting in line Time spent waiting to talk to a customer service agent Number of calories in a chocolate bar

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Chapter 5 - OPRE 3360

Probability distribution: is a table, formula, or graph that describes the values of a random variable and the probability associated with these values. Since we are describing a random variable (which can be discrete or continuous), we have two types of probability distributions: Discrete Probability Distribution, (this chapter) Continuous Probability Distribution (Chapter 8) Example: We asked 20 people how many siblings they have and recorded the reults in the following table: X: Number of Siblings Number of siblings 0 1 2 3 4

No. of respondents 3 6 5 4 2

Probability

Probability Notation: An upper-case letter will represent the name of the random variable, usually X. Its lower-case counterpart will represent the value of the random variable. The probability that the random variable X will equal x is: P (X = x) or more simply P (x)

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Chapter 5 - OPRE 3360 Two required conditions for a discrete probability function are as follow:

0 ≤ P (x) ≤ 1 X

P (x) = 1

all x

Example 7.1: Probability Distribution of Persons per Household The Statistical Abstract of the United States is published annually. One of the questions asks households to report the number of persons living in the household. The following table summarizes the data. Develop the probability distribution of the random variable defined as the number of persons per household. Number of persons 1 2 3 4 5 6 7 or more Total

Number of households (Millions) 31.1 38.6 18.8 16.2 7.2 2.7 1.4 116.0

P (x)

What is the probability that there are 4 or more persons in any given household?

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Chapter 5 - OPRE 3360 Probability Distributions and Populations The discrete probability distribution represents a population.

In Example 7.1, the distribution provided us with information about the population of numbers of persons per household.

And as we noted before, statistical inference deals with inference about populations. Since we have populations, we can describe them by computing various parameters. E.g. the population mean and population variance. Population Mean:

E(x) = µ = x1 P (x1 ) + x2 P (x2 ) + x3 P (x3 ) + · · · + xk P (xk ) =

k X

xi P (xi )

i=1

Population Variance:

V (x) = σ 2 = (x1 − µ)2 P (x1 ) + (x2 − µ)2 P (x2 ) + · · · + (xk − µ)2 P (xk ) =

k X i=1

(xi − µ)2 P (xi ) =

X

all x

x2 P (x) − µ2

Population Standard Deviation: σ=

√

σ2

Example 7.3: Describing the Population of the Number of Persons per Household Find the mean, variance, and standard deviation for the population of the number of persons per household (from Example 7.1). Assume that the category “7 or more” is actually 7.

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Now that we’ve introduced probability distributions in general, we need to introduce several specific probability distributions. Binomial probability distribution: The binomial distribution is the result of a binomial experiment, which has the following properties: 1. The experiment consists of a fixed number of trials. We represent the number of trials by n. 2. Each trial has two possible outcomes. We label one outcome a success, and the other a failure. 3. The probability of a success, denoted by p, does not change from trial to trial. The probability of failure is 1 − p. 4. The trials are independent which means that the outcome of one trial does not affect the outcomes of any other trials. If properties 2, 3, and 4 are satisfied, we say that each trial is a Bernoulli process. Adding property 1 yields the binomial experiment. Binomial random variable: The random variable of a binomial experiment is defined as the number of successes in the n trials. Example 1. flip a fair coin 10 times: 1. Fixed number of trials: n = 10 2. Each trial has two possible outcomes: {heads (success), tails (failure)} 3. P(success)= 0.50; P(failure)= 1 − 0.50 = 0.50 4. The trials are independent (i.e. the outcome of one coin flip cannot affect the outcomes of other flips).

Chapter 5 - OPRE 3360

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Example 2. A political survey asks 1,500 voters who they intend to vote for in an approaching election. In most elections in the United States, there are only two candidates, the Republican and Democratic nominees. 1. Fixed number of trials (voters): n = 1500 2. Each trial has two possible outcomes: the Republican and Democratic 3. The chance of selecting Democrat and Republican can be defined. 4. The trials are independent because the choice of one voter does not affect the choice of other voters. Binomial Probability Distribution: We want to find a probability function for a binomial random variable. Lets consider an example: flip a coin three times, chance of heads is 0.6, chance of tails is 0.4. This is a binomial experiment with three trails:

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Chapter 5 - OPRE 3360 Binomial Probability Distribution:   n! n x p (1 − p)n−x = f (x) = px (1 − p)n−x , for x = 0, 1, 2, ..., n x x!(n − x)!

Excel function: =BINOMDIST(#Successes, # trials, P(success), True (Cumulative) or False (not cumulative)) Expected value: E(x) = µ = np Variance: V ar(x) = σ 2 = np(1 − p) Standard deviation: σ =

p

np(1 − p)

Example 7.2: Probability Distribution of the Number of Sales A mutual fund salesperson has arranged to call on three people tomorrow. Based on past experience the salesperson knows that there is a 20% chance of closing a sale on each call. Determine the probability distribution of the number of sales the salesperson will make.

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Chapter 5 - OPRE 3360 Example - Evans Electronics:

Evans Electronics is concerned about a low retention

rate for its employees. In recent years, management has seen a turnover of 10% of the hourly employees annually. Thus, for any hourly employee chosen at random, management estimates a probability of 0.1 that the person will not be with the company next year. Choosing 3 hourly employees at random, what is the probability that 1 of them will leave the company this year? Calculate the expected value, variance and the standard deviation of the number of employee that will not be with the company next year.

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Chapter 5 - OPRE 3360 Practice Problems:

Question 1: For each of the following random variables, determine whether the random variable is continuous or discrete. (a) The time taken by a mechanic to repair an automobile. (b) The number of bags screened at the DFW airport on a given day. (c) The number of cracks on the surface of a newly molded part. (d) The time interval between the arrivals of two successive customers to a coffee shop. (e) The number of customers arriving at a coffee shop during a single day. (f) The time until a projectile returns to earth. (g) The volume of gasoline that is lost to evaporation during the filling of a gas tank. (h) The number of molecules in a sample of gas. (i) The number of defects in a manufactured part. (j) The diameter of a manufactured part.

Question 2: Consider the probability function f (x) = 0.01x3 , x = 1, 2, 3, 4.

(a) Verify that f is a valid function. (b) Compute P (X = 3). (c) Compute P (X ≥ 2).

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(d) Compute the expected value and variance of X . Compute the expected value and variance of X .

Question 3:

Suppose that 28% of students who take OPRE 3360 want a new copy of

the textbook, and the remaining students want a used copy. Suppose OPRE 3360 students begin arriving at the bookstore at some point in time. (a) On average, how many of the first 25 students want a new copy of the book? (b) What is the standard deviation of the number of new books wanted among the first 25 students? (c) What is the probability that exactly 3 of the first 11 students want a new book? (d) Determine the probability that less than 3 of the first 10 students want a new book. (e) Determine the probability that at least 2 of the first 3 students want a new book. Question 4: Suppose X is a binomial random variable with p = 0.55 and n = 9. Compute P (X = 2), P (X = 7), P (X < 2), P (X ≤ 2), E(X ), V (X )....