Discrete Probability Distributions

Introduction to Business Statistics, 5e

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(c)2000 South-Western College Publishing

Random Variable

A Random Variable is a function that assigns a numerical value to each outcome of an experiment.

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Discrete Random Variable

A discrete random variable is a random variable whose values are counting numbers or discrete data.

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Continuous Random Variable

A continuous random variable is a random variable for which any value is possible over some continuous range of values.

Introduction to Business Statistics, 5e

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(c)2000 South-Western College Publishing

Example 5.2

Consider a discrete random variable X havingpossible values of 1, 2, or 3. The corresponding probability for each value is:

1 with probability X = 2 with probability

3 with probability

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Example 5.2

Consider the function: P(X = x) = P(x) = x/6

or

X P(X)

1 2

3 1Introduction to Business Statistics, 5e

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Mean ofDiscrete Random Variables

The mean of a discrete random variable represents the average value of the random variable if you were to observe this variable over an indefinite period of time.

The mean of a discrete random variable is written as .

x P( x )Introduction to Business Statistics, 5e

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(c)2000 South-Western College Publishing

Variance ofDiscrete Random Variable

The variance of a discrete random variable, X, is a parameter describing the variation of the corresponding population.

The symbol used is 2 .

(x– )2P (x)Introduction to Business Statistics, 5e

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(c)2000 South-Western College Publishing

Discrete UniformRandom Variable

A discrete uniform random variable has the property that it is discrete and that its values all have the same probability of occurring.

a b

22

(b – a)( b– a 2)

12

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Binomial Random Variable

• The experiment consists of n repetitions, called trials.• Each trial has two mutually exclusive possible

outcomes, referred to as success and failure.• The n trials are independent.• The probability for a success for each trial is denoted

p; and remains the same for each trial• The random variable X is the number of successes out

of n trials.Introduction to Business Statistics, 5e

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(c)2000 South-Western College Publishing

Mean and Variance of a Binomial Random Variable

np(1– p)

np

Introduction to Business Statistics, 5e

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Using the Binomial Table to determine Binomial Probabilities

The binomial PMFs have been tabulated inTable A.1 for various values of n and p.

If n = 4 and p = 0.3 and you wish to find the P(2) locate n = 4 and x=2. Go across to p = 0.3and you will find the corresponding probability (after inserting the decimal in front of thenumber). This probability is 0.265.Introduction to Business Statistics, 5e

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(c)2000 South-Western College Publishing

Hypergeometric Distribution

• The hypergeometric distribution bears a strong resemblance to the binomial random variable.

• The experiment consists of n trials

• Has two possible outcomes

• Primary distinction between the Hypergeometric and the Binomial is that the trials in the Hypergeometric are not independent.

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Conditions for a Hypergeometric Distribution

• Population size = Nk members are S (successes) and N-k are F(failures)

• Sample size = n trialsobtained without replacement

• X = the number of successes

Introduction to Business Statistics, 5e

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(c)2000 South-Western College Publishing

Mean and Variance for theHypergeometric Distribution

xP( x) nkN

2 x2 P(x) – 2

n kN

1– k

N

N–nN–1

Introduction to Business Statistics, 5e

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(c)2000 South-Western College Publishing

Hypergeometric Probabilities

P(x) kCx N–kCn–x

NCn

Introduction to Business Statistics, 5e

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(c)2000 South-Western College Publishing

The Poisson Distribution

The poisson distribution is useful for counting the number of times a particular event occurs over a specified period of time or over a specified area.

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Conditions for the Poisson Distribution

• The Number of Occurrences in One Measurement Unit are independent of the Number of Occurrences in any other other Non-Overlapping Measurement Unit.

• The Expected Number of Occurrences in any given Measurement Unit are proportional to the size of the Measurement Unit.

• Events can not occur at exactly the same point in the Measurement Unit.

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Examples of Poisson Measurement Units

• Time:– Arrivals of customers at a service facility– Requests for replacement parts

• Linear:– Defects in linear feet of a spool of wire– Defects in square yards of carpet

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing

Poisson Probability Mass Function

P(x)x e –

x!

for x 0, 1, 2, 3, . . .Introduction to Business Statistics, 5e

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(c)2000 South-Western College Publishing

Mean and Variance of the Poisson Distribution

xP (x)Mean of X

(x – )2P(x)Variance of X

Introduction to Business Statistics, 5e

Kvanli/Guynes/Pavur

(c)2000 South-Western College Publishing