Discrete probabilityBusiness Statistics (BUSA 3101)

Dr. Lari H. Arjomandlariarjomand@clayton.edu

Discrete ProbabilityDistributions

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Random Variables Discrete Probability Distributions Expected Value and Variance Binomial Distribution Poisson Distribution (Optional Reading) Hypergeometric Distribution (Optional

Reading)

Random Variables

1. A random variable is a numerical description of the outcome of an experiment.

2. A discrete random variable may assume either a finite number of values or an infinite sequence of values.

3. A continuous random variable may assume any numerical value in an interval or Cllection of intervals.

Let x = number of TVs sold at the store in one day, where x can take on 5 values (0, 1, 2, 3, 4)

Let x = number of TVs sold at the store in one day, where x can take on 5 values (0, 1, 2, 3, 4)

Example: JSL Appliances

Discrete random variable with a finite number of values

Let x = number of customers arriving in one day, where x can take on the values 0, 1, 2, . . .

Let x = number of customers arriving in one day, where x can take on the values 0, 1, 2, . . .

Example: JSL Appliances

Discrete random variable with an infinite sequence of values

We can count the customers arriving, but there is nofinite upper limit on the number that might arrive.

Random VariablesExamples

Question Random Variable x Type

Familysize

x = Number of dependents reported on tax return

Discrete

Distance fromhome to store

x = Distance in miles from home to the store site

Continuous

Own dogor cat

x = 1 if own no pet; = 2 if own dog(s) only; = 3 if own cat(s) only; = 4 if own dog(s) and cat(s)

Discrete

Random VariablesDefinition & ExampleRandom VariablesDefinition & Example

Definition: A random variable is a quantity resulting from a random experiment that, by chance, can assume different values.

Example: Consider a random experiment in which a coin is tossed three times. Let X be the number of heads. Let H represent the outcome of a head and T the outcome of a tail.

The sample space for such an experiment will be: TTT, TTH, THT, THH, HTT, HTH, HHT, HHH.

Thus the possible values of X (number of heads) are X = 0, 1, 2, 3.

This association is shown in the next slide. Note: In this experiment, there are 8

possible outcomes in the sample space. Since they are all equally likely to occur, each outcome has a probability of 1/8 of occurring.

Example (Continued)Example (Continued)

TTT

TTH

THT

THH

HTT

HTH

HHT

HHH

TTT

TTH

THT

THH

HTT

HTH

HHT

HHH

0

1

1

2

1

2

2

3

0

1

1

2

1

2

2

3SampleSpace

X

Example (Continued)Example (Continued)

The outcome of zero heads occurred only once. The outcome of one head occurred three times. The outcome of two heads occurred three times. The outcome of three heads occurred only once. From the definition of a random variable, X as

defined in this experiment, is a random variable. X values are determined by the outcomes of the

experiment.

Example (Continued)Example (Continued)

Probability Distribution: DefinitionProbability Distribution: Definition

Definition: A probability distribution is a listing of all the outcomes of an experiment and their associated probabilities.

The probability distribution for the random variable X (number of heads) in tossing a coin three times is shown next.

Probability Distribution for Three Tosses of a CoinProbability Distribution for Three Tosses of a Coin

Data

Numerical(Quantitative)

Categorical(Qualitative)

Discrete Continuous

Data TypesData Types

Discrete Random Variable ExamplesDiscrete Random Variable Examples

Experiment RandomVariable

PossibleValues

Make 100 sales calls # Sales 0, 1, 2, ..., 100

Inspect 70 radios # Defective 0, 1, 2, ..., 70

Answer 33 questions # Correct 0, 1, 2, ..., 33

Count cars at tollbetween 11:00 & 1:00

# Carsarriving

0, 1, 2, ...,

We can describe a discrete probability distribution with a table, graph, or equation.

We can describe a discrete probability distribution with a table, graph, or equation.

Discrete Probability Distributions

The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable.

Discrete Probability Distributions

f(x) > 0f(x) > 0

f(x) = 1f(x) = 1

P(X) ≥ 0ΣP(X) = 1

The probability distribution is defined by a probability function, denoted by f(x),

which provides the probability for each value of the

random variable. The required conditions for a discrete

probability function are:

a tabular representation of the probability distribution for TV sales was developed.

Using past data on TV sales, …

Number Units Sold of Days

0 80 1 50 2 40 3 10 4 20

200

x f(x) 0 .40 1 .25 2 .20 3 .05 4 .10 1.00

80/200

Discrete Probability DistributionsExample

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0 1 2 3 40 1 2 3 4Values of Random Variable x (TV sales)Values of Random Variable x (TV sales)

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Discrete Probability Distributions

Graphical Representation of Probability Distribution

As we said, the probability distribution of a discrete random variable is a table, graph, or formula that gives the probability associated with each possible value that the variable can assume.

Example : Number of Radios Sold at Sound City in a Weekx, Radios p(x), Probability 0 p(0) = 0.03 1 p(1) = 0.20 2 p(2) = 0.50 3 p(3) = 0.20 4 p(4) = 0.05 5 p(5) = 0.02

Expected Value of a Discrete Random Variable

The mean or expected value of a discrete random

variable is:xAll

X xxp )(

Example: Expected Number of Radios Sold in a Weekx, Radios p(x), Probability x p(x) 0 p(0) = 0.03 0(0.03) = 0.00 1 p(1) = 0.20 1(0.20) = 0.20 2 p(2) = 0.50 2(0.50) = 1.00 3 p(3) = 0.20 3(0.20) = 0.60 4 p(4) = 0.05 4(0.05) = 0.20 5 p(5) = 0.02 5(0.02) = 0.10

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