1. Discrete Probability Distributions
Statistics
Discrete Probability Distributions

2. Contents Random Variables Discrete Probability Distributions
Expected Value and Variance
Binomial Distribution
.10
.20
.30
.40
Poisson Distribution
Hypergeometric Distribution

3. STATISTICS in PRACTICE
Citibank makes available a wide range of financial services.
Citibanking’s automatic teller
machines (ATMs) located in
Citicard Banking Centers
(CBCs), let customers do all
their banking in one place with
the touch of a finger.

4. STATISTICS in PRACTICE
Periodic CBC capacity studies are
used to analyze customer waiting times and to determine whether additional ATMs are needed.
Data collected by Citibank showed that the random customer arrivals followed a probability distribution known as the Poisson distribution.

5. Random Variables A random variable is a numerical
description of the outcome of an experiment.
A discrete random variable may assume
either a finite number of values or an infinite
sequence of values.
A continuous random variable may assume
any numerical value in an interval or
collection of intervals.

6. Differences between outcomes and random variables
Example: Tossing a dice
Possible outcomes: 1, 2, 3, 4, 5, and 6
One can define random variables as
1 if outcome is greater than 3, and
0 if outcome is smaller or equal to 3. Or
1 if outcome is odd numbers
0 if outcome is even numbers

7. Discrete Random Variables
Example 1.
The certified public accountant (CPA)
examination has four parts.
Define a random variable as x = the number of
parts of the CPA examination passed and It is
a discrete random variable because it may
assume the finite number of values 0, 1, 2, 3, or 4.

8. Discrete Random Variables
Example 2.
An experiment of cars arriving at a tollbooth. The random variable is x = the number of cars arriving during a one-day period. The possible values for x come from the sequence of integers 0, 1, 2, and so on. x is a discrete random variable assuming one of the values in this infinite sequence.

9. Discrete Random Variables
Examples of Discrete Random Variables

10. Example: JSL Appliances
Discrete random variable with a finite
number of values
Let x = number of TVs sold at the store in
one day, where x can take on 5 values
(0, 1, 2, 3, 4)

11. 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.

12. Random Variables Question Random Variable x Type Family size
x = Number of dependents
reported on tax return
Discrete
Distance from
home to store
x = Distance in miles from
home to the store site
Continuous
Own dog
or 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

13. Continuous Random Variables
Example 1.
Experimental outcomes based on measurement scales such as time, weight, distance, and temperature can be described by continuous random variables.

14. Continuous Random Variables
Example 2.
An experiment of monitoring incoming telephone calls to the claims office of a major insurance company. Suppose the random variable of interest is x = the time between consecutive incoming calls in minutes. This random variable may assume any value in the interval x ≥ 0.

15. Continuous Random Variables
Example of Continuous Random Variables

16. Discrete Probability Distributions
The probability distribution for a random
variable describes how probabilities are
distributed over the values of the random
variable.
We can describe a discrete probability
distribution with a table, graph, or equation.

17. 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) > 0
f(x) = 1

18. Discrete Probability Distributions
Example: Probability Distribution for the Number of Automobiles Sold During a Day at Dicarlo Motors.

19. Discrete Probability Distributions
Using past data on TV sales, …
a tabular representation of the probability
distribution for TV sales was developed.
Number
Units Sold of Days
200
x f(x)
1.00
80/200

20. Discrete Probability Distributions
Graphical Representation of Probability Distribution
.10
.20
.30
.40
.50
Probability
Values of Random Variable x (TV sales)

21. Discrete Uniform Probability Distribution
The discrete uniform probability distribution
is the simplest example of a discrete
probability distribution given by a formula.

22. Discrete Uniform Probability Distribution
The discrete uniform probability function is
the values of the
random variable
are equally likely
f(x) = 1/n
where:
n = the number of values the random
variable may assume

23. Discrete Uniform Probability Distribution
Example:
An experiment of rolling a die we define the random variable x to be the number of dots on the upward face. There are n = 6 possible values for the random variable;
x = 1, 2, 3, 4, 5, 6. The probability function for this discrete uniform random variable is f (x) = 1/6 x = 1, 2, 3, 4, 5, 6.

24. Discrete Uniform Probability Distribution
x f (x) /6 /6 /6 /6 /6 /6

25. Discrete Uniform Probability Distribution
Example:
Consider the random variable x with the following discrete probability distribution.
This probability distribution can be defined by the formula f (x) = x/ 10 for x = 1, 2, 3, or 4.
x f (x)
/10
/10
/10
/10

26. Expected Value and Variance
The expected value, or mean, of a random
variable is a measure of its central location.
E(x) =  = x f(x)
The variance summarizes the variability in the
values of a random variable.
Var(x) =  2 = (x - )2f(x)

27. Expected Value and Variance
The standard deviation,  , is defined
as the positive square root of the variance.
Here, the expected value and variance
are computed from random variables
instead of outcomes

28. Expected Value and Variance
Example: Calculation of the Expected Value for the Number of Automobiles Sold During A Day at Dicarlo Motors.

29. Expected Value and Variance
Example: Calculation of the Variance for
the Number of Automobiles Sold During A
Day at Dicarlo Motors.
The standard deviation is

30. Expected Value and Variance
x f(x) xf(x)
E(x) =
expected number of TVs sold in a day

31. Expected Value and Variance
Variance and Standard Deviation x
x - 
(x - )2
f(x)
(x - )2f(x) 1 2 3 4
-1.2
-0.2
0.8
1.8
2.8
1.44
0.04
0.64
3.24
7.84
.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 = TVs

32. Binomial Distribution
Four Properties of a Binomial Experiment
1. The experiment consists of a sequence of n
identical trials.
2. Two outcomes, success and failure, are possible
on each trial.
3. The probability of a success, denoted by p, does
not change from trial to trial.
stationarity
assumption
4. The trials are independent.

33. Binomial Distribution
Our interest is in the number of successes
occurring in the n trials.
We let x denote the number of successes
occurring in the n trials.

34. Binomial Distribution
Binomial Probability Function
where:
f(x) = the probability of x successes in
n trials,
n = the number of trials,
p = the probability of success on any
one trial.

35. Binomial Distribution
Number of experimental
outcomes providing exactly
x successes in n trials =
Probability of a particular
sequence of trial outcomes
with x successes in n trials =

36. Binomial Distribution
Binomial Probability Function

37. Binomial Distribution
Example:
The experiment of tossing a coin five times
and on each toss observing whether the coin lands with a head or a tail on its
upward face. we want to count the number
of heads appearing over the five tosses.
Does this experiment show the properties of a binomial experiment?

38. Binomial Distribution
Note that:
1. The experiment consists of five identical
trials; each trial involves the tossing of one
coin.
2. Two outcomes are possible for each trial: a
head or a tail. We can designate head a
success and tail a failure.

39. Binomial Distribution
Note that:
3. The probability of a head and the probability of
a tail are the same for each trial, with p = .5
and 1- p = .5.
4. The trials or tosses are independent because the
outcome on any one trial is not affected by
what happens on other trials or tosses.

40. Binomial Distribution
Example: Evans Electronics
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.

41. Binomial Distribution
Using the Binomial Probability Function
Choosing 3 hourly employees at random, what is the probability that 1 of them will leave the company this year?
Let: p = .10, n = 3, x = 1

42. Binomial Distribution
Tree Diagram
1st Worker
2nd Worker
3rd Worker x
Prob.
L (.1) 3
.0010
Leaves (.1) 2
.0090
S (.9)
Leaves
(.1)
L (.1)
.0090 2
Stays (.9) 1
.0810
S (.9)
L (.1) 2
.0090
Leaves (.1)
Stays
(.9)
S (.9) 1
.0810
L (.1) 1
.0810
Stays (.9)
.7290
S (.9)

43. Binomial Distribution
Using Tables of Binomial Probabilities

44. Binomial Distribution
Expected Value
E(x) =  = np
Variance
Var(x) =  2 = np(1 - p)
Standard Deviation

45. Binomial Distribution
Expected Value
E(x) =  = 3(.1) = .3 employees out of 3
Variance
Var(x) =  2 = 3(.1)(.9) = .27
Standard Deviation

46. Poisson Distribution A Poisson distributed random variable is often
useful in estimating the number of occurrences
over a specified interval of time or space
It is a discrete random variable that may
assume an infinite sequence of values
(x = 0, 1, 2, ).

47. Poisson Distribution
Examples of a Poisson distributed random variable:
the number of knotholes in 14 linear feet of
pine board
the number of vehicles arriving at a
toll booth in one hour

48. Poisson Distribution Two Properties of a Poisson Experiment
The probability of an occurrence is the
same for any two intervals of equal length.
The occurrence or nonoccurrence in any
interval is independent of the occurrence
or nonoccurrence in any other interval.

49. Poisson Distribution Poisson Probability Function where:
f(x) = probability of x occurrences in
an interval,
 = mean number of occurrences in
e =

50. Poisson Distribution Example: We are interested in the number of
arrivals at the drive-up teller window of a
bank during a 15-minute period on weekday
mornings. Assume that the probability of a
car arriving is the same for any two time
periods of equal length and that the arrival or
nonarrival of a car in any time period is
independent of the arrival or nonarrival in any
other time period.

51. Poisson Distribution The Poisson probability function is applicable.
Suppose that the average number of cars arriving
in a 15-minute period of time is 10; in this case,
the following probability function applies.
The random variable here is
x = number of cars arriving in any 15-minute period.

52. Poisson Distribution Example: Mercy Hospital
Patients arrive at the emergency
room of Mercy Hospital at the average rate of
6 per hour on weekend evenings.
What is the probability of 4 arrivals in 30 minutes on a weekend evening?

53. Poisson Distribution Using the Poisson Probability Function
MERCY
Poisson Distribution
Using the Poisson Probability Function
 = 6/hour = 3/half-hour, x = 4

54. MERCY
Poisson Distribution
Using Poisson Probability Tables

55. Poisson Distribution Poisson Distribution of Arrivals actually,
MERCY
Poisson Distribution
Poisson Distribution of Arrivals
Poisson Probabilities
0.00
0.05
0.10
0.15
0.20
0.25 1 2 3 4 5 6 7 8 9 10
Number of Arrivals in 30 Minutes
Probability
actually,
the sequence
continues:
11, 12, …

56. Poisson Distribution A property of the Poisson distribution is that
the mean and variance are equal.
m = s 2

57. Poisson Distribution Variance for Number of Arrivals
MERCY
Poisson Distribution
Variance for Number of Arrivals
During 30-Minute Periods
m = s 2 = 3

58. Hypergeometric Distribution
The hypergeometric distribution is closely related
to the binomial distribution.
However, for the hypergeometric distribution:
the trials are not independent, and
the probability of success changes from trial
to trial.

59. Hypergeometric Distribution
Hypergeometric Probability Function
for 0 < x < r
where: f(x) = probability of x successes in n trials,
n = number of trials,
N = number of elements in the population,
r = number of elements in the population
labeled success.

60. Hypergeometric Distribution
Hypergeometric Probability Function
for 0 < x < r

61. Hypergeometric Distribution
Hypergeometric Probability Function
number of ways x successes can be
selected from a total of r successes
in the population =
number of ways n – x failures can be selected from a total of N – r failures
in the population =
number of ways
a sample of size n can be selected
from a population of size N =

62. Hypergeometric Distribution
Example: A Quality Control Application.
Electric fuses produced by Ontario Electric are packaged in boxes of 12 units each. Suppose an inspector randomly selects 3 of the 12 fuses in a box for testing.
If the box contains exactly 5 defective fuses, what is the probability that the inspector will find exactly 1 of the 3 fuses defective?

63. Hypergeometric Distribution
In this application, n = 3 and N = 12.
With r = 5 defective fuses in the box.

64. Hypergeometric Distribution
The probability of finding x = 1 defective fuse is
What is the probability of finding at least 1 defective fuse?

65. Hypergeometric Distribution
The probability of x = 0 is
we conclude that the probability of finding at least 1 defective fuse must be =

66. Hypergeometric Distribution
Example: Neveready
Bob Neveready has removed two
dead batteries from a flashlight
and inadvertently mingled
them with the two good
batteries he intended
as replacements. The four batteries look identical. Bob now randomly selects two of the four batteries. What is the probability he selects the two good batteries?
ZAP

67. Hypergeometric Distribution
Using the Hypergeometric Function
where:
x = 2 = number of good batteries selected
n = 2 = number of batteries selected
N = 4 = number of batteries in total
r = 2 = number of good batteries in total

68. Hypergeometric Distribution
Mean
Variance

69. Hypergeometric Distribution
Mean
Variance

70. Hypergeometric Distribution
Consider a hypergeometric distribution with
n trials and let p = (r/n) denote the probability
of a success on the first trial.
If the population size is large, the term
(N – n)/(N – 1) approaches 1.

71. Hypergeometric Distribution
The expected value and variance can be written
E(x) = np and Var(x) = np(1 – p).
Note that these are the expressions for the
expected value and variance of a binomial
distribution.
continued

72. Hypergeometric Distribution
When the population size is large, a
hypergeometric distribution can be approximated
by a binomial distribution with n trials and a
probability of success p = (r/N).