Discrete Probability Distributions

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Discrete Probability Distributions

• Random variables
• Discrete probability distributions
• Expected value and variance
• Binomial probability distribution

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Random Variables
A random variable is a numerical description of
the outcome of an experiment
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Discrete Random Variables
A discrete random variable may assume a finite
number of numerical values or an infinite
sequence of values such as 0, 1, 2, . . .
Example 1 passed driving test 2 failed
driving test
This is a discrete variable because it is either
pass or failyou cant score 1½, for example.
Notice also it assumes a finite number of values
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Example JSL Appliances

• Discrete random variable with an infinite
  sequence of values

Let x number of customers arriving in one
day, where x can take on the values 0, 1, 2, .
. .
We can count the customers arriving, but there
is no finite upper limit on the number that might
arrive.
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Continuous Random Variables
Continuous random variables can assume an
infinite number of values within a defined
interval.
The liquid in these bottles (x) must be between 0
and 32 ounces. But x could be 2 oz., 2.1 oz.,
2.01 oz, 2.001 oz., . . .
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Random Variables
Type
Random Variable x
Question
Family size
x Number of dependents in family reported on
tax return
Discrete
x Distance in miles from home to the store site
Distance from home to store
Continuous
Own dog or cat
Discrete
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)
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Discrete Probability Distributions
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
f(x) gt 0
?f(x) 1
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Example JSL Appliances

• Using past data on TV sales,

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

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
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Example JSL Appliances

• Graphical Representation of the Probability
  Distribution

Probability
0 1 2 3 4
Values of Random Variable x (TV sales)
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Discrete Uniform Probability Distribution
This is the simplest probability distribution
described by a formula. It assumes that possible
values of random variables are equally likely
n number of values the random variable may
assume.
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Example Rolling a Die
x f(x)
1 1/6
2 1/6
3 1/6
4 1/6
5 1/6
6 1/6
Note that n 6
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Expected Value E(x)
The expected value, or mean, of a random
variable, is a measure of central location for
the random variable
For a discrete random variable x we have
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Notice that this is a weighted average of the
values a random variable can assume. The
weights are the probabilities.
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Example JSL Appliances

• Expected Value of a Discrete Random Variable

x f(x) xf(x) 0 .40
.00 1 .25 .25 2 .20
.40 3 .05 .15 4 .10
.40 E(x) 1.20
expected number of TVs sold in a day
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Variance of a Discrete Random Variable
The variance of a random variable x is a
weighted average of the squared deviations of a
random variable from its mean (expected) value.
The weights are the probabilities.
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Standard Deviation of a Random Variable (s)
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Example JSL Appliances

• Variance and Standard Deviation
• of a Discrete Random Variable

(x - ?)2
f(x)
(x - ?)2f(x)
x
x - ?
-1.2 -0.2 0.8 1.8 2.8
1.44 0.04 0.64 3.24 7.84
0 1 2 3 4
.40 .25 .20 .05 .10
.576 .010 .128 .162 .784
TVs squared
Variance of daily sales s 2 1.660
Standard deviation of daily sales 1.2884 TVs
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Using Excel to Compute the Expected Value,
Variance, and Standard Deviation

• Formula Worksheet

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Using Excel to Compute the Expected Value,
Variance, and Standard Deviation

• Value Worksheet

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The Binomial Distribution
This is a very useful tool for multi-step
experiments where each step has 2 outcomeshence
the term binomial.
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Properties of Binomial Experiment

1. The experiment consists of a sequence of n
   identical trials.
2. Two outcomes are possible on each trial. We refer
   to one outcome as a success and the other outcome
   as a failure.
3. The probability of success, denoted by ?, does
   not change from trial to trial. Consequently, the
   probability of failure, denoted by 1 ? , does
   not change from trial to trial.
4. The trials are independent.

Stationarity assumption
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We are interested in computing the number of
successes (x) for n number of trials
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Binomial Probability Distribution
The binomial distribution is given by
Where f(x) probability of success in n
trials n number of trials p probability of
success in any one trial.
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Example Evans Electronics

• Binomial Probability Distribution
• Evans is concerned about a low retention rate
  for 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.

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Evans electronics
If we selected three (3) employees at random,
what is the probability that one(1) will leave
the company within the year?

• Notice that
• The experiment has three identical trialsthat
  is, n 3.
• There are two outcomes for each trialthe
  employee leaves (S) or the employee stays (F).
• The probability that an employee will leave is
  .1that is, ? .1
• The decision of each employee to leave is
  independent of the decisions made by the other
  employees.

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Counting the Number of Outcomes
FirstEmployee
SecondEmployee
ThirdEmployee
ExperimentalOutcome
Value of x32212110
(S,S,S)
S
S
F
(S,S,F)
S
F
(S,F,S)
S
F
(S,F,F)
S
S
F
(F,S,S)
(F,S,F)
F
F
S
(F,F,S)
(F,F,F)
F
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Number of Experimental Outcomes Providing Exactly
x Successes in n trials
In our Evans Electronics example, n 3 and x
1. Thus
Refer to the tree diagram to verify this is right
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What is the probability the first employee
selected will leave and second and third will
stay? Note this is outcome (S, F, F)
Because these events are independent, we can
multiply probabilities. . Thus the probability of
(S, S, F) is given by
Thus we have
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Trial Outcomes
Experimental Outcome Probability of Experimental Outcome
(S,F,F)
(F,S,F)
(F,F,S)
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Binomial Probability Distribution

• Binomial Probability Function

Probability of a particular sequence of trial
outcomes with x successes in n trials
Number of experimental outcomes providing
exactly x successes in n trials
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Probability Distribution for the Number of
Employees Leaving Within the Year
x f(x)
0
1
2
3
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Using Excel to ComputeBinomial Probabilities

• Formula Worksheet

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Using Excel to ComputeBinomial Probabilities

• Value Worksheet

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Using Excel to ComputeCumulative Binomial
Probabilities

• Formula Worksheet

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Using Excel to ComputeCumulative Binomial
Probabilities

• Value Worksheet

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Expected Value and Variance for a Binominal
Distribution
The expected value is computed by
The variance is computed by
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Evans Electronics Example
Remember that n 3 and ? .1. Thus
Note also that