Slide 2 Econ10/Mgt 10 Stuffler 1 Discrete Random Variables
Discrete Probability Distributions Binomial Distribution Poisson
Distribution Hypergeometric Distribution Slide 3 Econ10/Mgt 10
Stuffler 2 Discrete Probability Distributions Discrete Random
Variables Sample Space includes all mutually exclusive outcomes
Probabilities from subjective, frequency or subjective methods Two
conditions apply: Slide 4 Econ10/Mgt 10 Stuffler 3 Probability
distributions can be estimated from relative frequencies. Consider
the discrete (countable) number of televisions per household
(millions) from US survey data: Discrete Probability Distributions
Slide 5 Econ10/Mgt 10 Stuffler 4 Discrete Probability Distributions
Probability distributions can be estimated from relative
frequencies. Consider the discrete (countable) number of
televisions per household from US survey data: 1,218 101,501 =
0.012 Slide 6 Econ10/Mgt 10 Stuffler 5 Discrete Probability
Distributions Probability distributions can be estimated from
relative frequencies. Consider the discrete (countable) number of
televisions per household from US survey data: EX: P( X =4) = P(4)
= 0.076 = 7.6% Slide 7 Econ10/Mgt 10 Stuffler 6 Discrete
Probability Distributions What is the probability there is at least
one television but no more than three in any given household? Slide
8 Econ10/Mgt 10 Stuffler 7 Discrete Probability Distributions What
is the probability there is at least one television but no more
than three in any given household? Slide 9 Econ10/Mgt 10 Stuffler 8
Discrete Probability Distributions What is the probability there is
at least one television but no more than three in any given
household? at least one television but no more than three P(1 X 3)
= P(1) + P(2) + P(3) =.319 +.374 +.191 =.884 Slide 10 Econ10/Mgt 10
Stuffler 9 Discrete Probability Distributions Assume a mutual fund
salesman knows that there is 20% chance of closing a sale on each
call he makes. Slide 11 Econ10/Mgt 10 Stuffler 10 Discrete
Probability Distributions What is the probability distribution of
the number of sales if he plans to call three customers? Let S
denote success, making a sale: P(S) =.20, then S , not making a
sale: P(S ) =.80 Slide 12 Econ10/Mgt 10 Stuffler 11 Discrete
Probability Distributions Developing a Probability Distribution
Tree P(S)=.2 P(S)=.8 Sales Call 1 Slide 13 Econ10/Mgt 10 Stuffler
12 Discrete Probability Distributions Developing a Probability
Distribution Tree P(S)=.2 P(S)=.8 P(S)=.2 P(S)=.8 P(S)=.2 Sales
Call 1Sales Call 2 Slide 14 Econ10/Mgt 10 Stuffler 13 Discrete
Probability Distributions Developing a Probability Distribution
Tree P(S)=.2 P(S)=.8 P(S)=.2 P(S)=.8 S S S P(S)=.2 P(S)=.8 P(S)=.2
Sales Call 1Sales Call 2Sales Call 3 Slide 15 Econ10/Mgt 10
Stuffler 14 Discrete Probability Distributions Developing a
Probability Distribution Tree P(S)=.2 P(S)=.8 P(S)=.2 P(S)=.8 S S S
P(S)=.2 P(S)=.8 P(S)=.2 Sales Call 1Sales Call 2Sales Call 3 Slide
16 Econ10/Mgt 10 Stuffler 15 Discrete Probability Distributions
Developing a Probability Distribution P(S)=.2 P(S)=.8 P(S)=.2
P(S)=.8 S S S P(S)=.2 )=.8 P(S)=.8 P(S)=.2 XP(x) 3.2 3 =.008
23(.032)=.096 13(.128)=.384 0.8 3 =.512 Sales Call 1Sales Call
2Sales Call 3 Slide 17 Econ10/Mgt 10 Stuffler 16 Discrete
Probability Distributions Explain how to derive.032 and.128 P(S)=.2
P(S C )=.8 P(S)=.2 P(S C )=.8 S S S S S S C S S C S S S C S C S C S
S S C S S C S C S C S S C S C S C P(S)=.2 P(S C )=.8 P(S)=.2 XP(x)
3.2 3 =.008 23(.032)=.096 13(.128)=.384 0.8 3 =.512 Sales Call
1Sales Call 2Sales Call 3 Slide 18 Econ10/Mgt 10 Stuffler 17
Discrete Probability Distributions The P(X=2) is: P(S)=.2 P(S C
)=.8 P(S)=.2 P(S C )=.8 S S S S S S C S S C S S S C S C S C S S S C
S S C S C S C S S C S C S C P(S)=.2 P(S C )=.8 P(S)=.2 XP(x) 3.2 3
=.008 23(.032)=.096 13(.128)=.384 0.8 3 =.512 (.2)(.2)(.8)=.032
Sales Call 1Sales Call 2Sales Call 3 Slide 19 Econ10/Mgt 10
Stuffler 18 Discrete Probability Distributions A discrete
probability distribution represents a population Example:
Population of number of TVs per household Example: Population of
sales call outcomes Since we have populations, we can describe them
by computing various parameters: Population Mean and Population
Variance Slide 20 Econ10/Mgt 10 Stuffler 19 Discrete Probability
Distributions Population Mean (Expected Value) - W eighted average
of all values with the weights being the probabilities - Expected
value of X, E(X) Slide 21 Econ10/Mgt 10 Stuffler 20 Discrete
Probability Distributions Population variance - weighted average of
the squared deviations from the mean. Short cut formula for the
variance Standard deviation formula Slide 22 Econ10/Mgt 10 Stuffler
21 Discrete Probability Distributions Find the mean, variance, and
standard deviation for the population of the number of color
televisions per household. = 0(.012) + 1(.319) + 2(.374) + 3(.191)
+ 4(.076) + 5(.028) = 2.084 Slide 23 Econ10/Mgt 10 Stuffler 22
Discrete Probability Distributions Find the mean, variance, and
standard deviation for the population of the number of color
televisions per household. = (0 2.084) 2 (.012) + (1 2.084) 2
(.319)++(5 2.084) 2 (.028) = 1.107 Slide 24 Econ10/Mgt 10 Stuffler
23 Discrete Probability Distributions Find the mean, variance, and
standard deviation for the population of the number of color
televisions per household. = 1.052 Slide 25 Econ10/Mgt 10 Stuffler
24 Discrete Probability Distributions Special application of
Expected Value Suppose the probability that an insurance agent
makes a sale is.20 and after costs earns a commission of $525. If
he/she does not make a sale, they must pay $75 in costs. What is
their expected value from a sales call? Does the benefit exceed the
cost or is the opposite true? Slide 26 Econ10/Mgt 10 Stuffler 25
Discrete Probability Distributions Special application of Expected
Value Let X be the discrete random variable of making a sale call x
P(x) xP(x) Sale $ 525.20 $ 105 No Sale - $ 75.80 - $ 60 E(x) = =
xP(x) = $ 45 Slide 27 Econ10/Mgt 10 Stuffler 26 Binomial
Distribution Binomial distribution is the probability distribution
that results from doing a binomial experiment which have the
properties: 1.Fixed number of identical trials, represented as n.
2.Each trial has two possible outcomes: a success or failure 3.For
all trials, the probability of success, P(success)=p, and the
probability of failure, P(failure)=1p=q, are constant. 4.The trials
are independent Slide 28 Econ10/Mgt 10 Stuffler 27 Several Binomial
Distributions Slide 29 Econ10/Mgt 10 Stuffler 28 Binomial
Distribution Success and failure: labels for binomial experiment
outcomes, no value judgment is implied. EX: Coin flip results in
either heads or tails. If we define heads as success, then tails is
considered a failure. Other binomial examples: An election
candidate wins or loses An employee is male or female A worker is
employed or unemployed Slide 30 Econ10/Mgt 10 Stuffler 29 Binomial
Distribution where x = number of successes in n trials, n x =
number of failures in n trials, p x = the probability of success
raised to the number of successes, and q n-x = probability of
failure raised to the number of failures Binomial Formula: Slide 31
Econ10/Mgt 10 Stuffler 30 Binomial Distribution The random variable
of a binomial experiment is defined as the number of successes in
the n trials, and is called the binomial random variable. EX: 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)=10.50 = 0.50 4) The trials are
independent (i.e. the outcome of heads on the first flip will have
no impact on subsequent coin flips). Flipping a coin ten times is a
binomial experiment since all conditions are met. Slide 32
Econ10/Mgt 10 Stuffler 31 Binomial Distribution Another sales call
example: Assume a mutual fund salesman knows that there is 20%
chance of closing a sale on each call he makes. We want to
determine the probability of making two sales in three calls:
P(sale) =.2 P(no sale) =.8 P(X=2) = 3! =.096 2!(3-2)!.2 2.8.8 Slide
33 Econ10/Mgt 10 Stuffler 32 Binomial Distribution Another sales
call example: Assume a mutual fund salesman knows that there is 20%
chance of closing a sale on each call he makes We want probability
of making two sales in three calls P(sale) =.2 P(no sale) =.8 GOOD
NEWS! MegastatProbabilityDiscrete DistributionsBinomial Slide 34
Econ10/Mgt 10 Stuffler 33 Binomial Distribution Slide 35 Econ10/Mgt
10 Stuffler 34 Pat Statsly Pat Statsly is a student (not a good
student) taking a statistics course. Pats exam strategy is to rely
on luck for the first test. The test consists of 10 multiple-choice
questions. Each question has five possible answers, only one of
which is correct. Pat plans to guess the answer to each question.
What is the probability that Pat gets no answers correct? What is
the probability that Pat gets two answers correct? Slide 36
Econ10/Mgt 10 Stuffler 35 Pat Statsly n=10 P(correct) = p = 1/5
=.20 P(wrong) = q =.80 Is this a binomial experiment? Check the
conditions Slide 37 Econ10/Mgt 10 Stuffler 36 Pat Statsly n=10
P(correct) = p = 1/5 =.20 P(wrong) = q =.80 Is this a binomial
experiment? Check the conditions: There is a fixed finite number of
trials (n=10). An answer can be either correct or incorrect. The
probability of a correct answer (P(success)=.20) does not change
from question to question. Each answer is independent of the
others. Slide 38 Econ10/Mgt 10 Stuffler 37 Pat Statsly n=10, and
P(success) =.20 What is the probability that Pat gets no answers
correct? EX: P(x=0) Whats the interpretation of this result? Slide
39 Econ10/Mgt 10 Stuffler 38 Pat Statsly n=10, and P(success) =.20
What is the probability that Pat gets no answers correct? EX:
P(x=0) Pat has about an 11% chance of getting no answers correct
using the guessing strategy. Slide 40 Econ10/Mgt 10 Stuffler 39 Pat
Statsly Slide 41 Econ10/Mgt 10 Stuffler 40 Pat Statsly n=10, and
P(success) =.20 What is the probability that Pat gets two answers
correct? EX: P(x=2) Slide 42 Econ10/Mgt 10 Stuffler 41 Pat Statsly
We have been using the binomial probability distribution to find
probabilities for individual values of x. To answer the question:
Find the probability that Pat fails the quiz requires a cumulative
probability, that is, P(X x) If a grade on the quiz is less than
50% (i.e. 5 questions out of 10), thats considered a failed quiz.
Slide 43 Econ10/Mgt 10 Stuffler 42 Pat Statsly We have been using
the binomial probability distribution to find probabilities for
individual values of x. To answer the question: Find the
probability that Pat fails the test requires a cumulative
probability, that is, P(X x) If a grade on the test is less than
50% (i.e., 5 questions out of 10), thats considered a failed test.
We want to know what is: P(X 4) to answer Slide 44 Econ10/Mgt 10
Stuffler 43 Pat Statsly P(X 4) = P(0) + P(1) + P(2) + P(3) + P(4)
=.9672 What is the interpretation of this result? Slide 45
Econ10/Mgt 10 Stuffler 44 Pat Statsly Its about 97% probable that
Pat will fail the test using the luck strategy and guessing at
answers. Slide 46 Econ10/Mgt 10 Stuffler 45 Binomial Table
Calculating binomial probabilities by hand is tedious and error
prone. There is an easier way. Refer to Table E-6 in the
Appendices. For the Pat Statsly example, n=10, so go to the n=10
table: Look in the Column, p=.20 and substitute into: P(X 4) = P(0)
+ P(1) + P(2) + P(3) + P(4) P(X 4) =.1074 +.2684 +.3020 +.2013
+.0881 =.9672 The probability of Pat failing the test is 96.72%
Slide 47 Econ10/Mgt 10 Stuffler 46 Poisson Distribution Named for
Simeon Poisson, Poisson distribution - Discrete probability
distribution There there are no trials Number of independent events
(successes) occurring in a fixed time period or region of space
that occur with a known average rate such as, arrivals, departures,
or accidents, number of baskets in a quarter, etc. Slide 48
Econ10/Mgt 10 Stuffler 47 Poisson Distribution For example: The
number of cars arriving at a service station in 1 hour. (The
interval of time is 1 hour.) The number of flaws in a bolt of
cloth. (The specific region is a bolt of cloth.) The number of
accidents in 1 day on a particular stretch of highway. (The
interval is defined by both time, 1 day, and space, the particular
stretch of highway.) Slide 49 Econ10/Mgt 10 Stuffler 48 Poisson
Distribution Poisson random variable - number of successes that
occur in a period of time or an interval of space EX: On average,
96 trucks arrive at a border crossing every hour. EX: Number of
typographical errors in a new textbook edition averages 1.5 per 100
pages. Slide 50 Econ10/Mgt 10 Stuffler 49 Poisson Distribution The
Poisson random variable is the number of successes that occur in a
period of time or an interval of space in a Poisson experiment. EX:
On average, 96 trucks arrive at a border crossing every hour. E.g.
The number of typographic errors in a new textbook edition averages
1.5 per 100 pages. successes time period successes (?!)interval
Slide 51 Econ10/Mgt 10 Stuffler 50 Poisson Distribution Poisson
experiment has four defining characteristics or properties: 1.The
number of successes that occur in any interval is independent of
the number of successes that occur in any other interval 2.The
probability of a success in an interval is the same for all
equal-size intervals 3.The probability of a success is proportional
to the size of the interval 4.Only one value is required to
determine the probability of a designated number of events
occurring during an interval Slide 52 Econ10/Mgt 10 Stuffler 51
Poisson Distribution The probability that a Poisson random variable
assumes a value of x is given by: = n(p) = number of successes
times the probability of success The unit of time or the interval
should be short enough so that the mean rate is not large ( <
20) YOUR TEXTBOOK USES (lambda) REPRESENT THE MEAN NUMBER OF
SUCCESSES IN THE INTERVAL Slide 53 Econ10/Mgt 10 Stuffler 52
Poisson Distribution EXAMPLE The number of typographical errors in
new editions of textbooks varies considerably from book to book.
After some analysis she concludes that the number of errors is
Poisson distributed with a mean of 1.5 per 100 pages. The
instructor randomly selects 100 pages of a new book. What is the
probability that there are no typos? MEGASTATPROBABILITYDISCRETE
DISTRIBUTIONSPOISSON AND ENTER THE MEAN VALUE: 1.5, CLICK OK Slide
54 Econ10/Mgt 10 Stuffler 53 Poisson Distribution Slide 55
Econ10/Mgt 10 Stuffler 54 Mean and Variance of a Poisson Random
Variable If x is a Poisson random variable with parameter, then
Mean number of events per unit of time or space = Variance = V(X) =
2 = Slide 56 Econ10/Mgt 10 Stuffler 55 Poisson Distribution As
mentioned on the first Poisson Distribution slide: The probability
of a success is proportional to the size of the interval Thus,
knowing an error rate of 1.5 typos per 100 pages, we can determine
a mean value for a 400 page book as: = n(p) = 1.5(4) = 6 typos /
400 pages. Slide 57 Econ10/Mgt 10 Stuffler 56 Poisson Distribution
For a 400 page book, what is the probability that there are no
typos? P(X=0) = Interpretation? Slide 58 Econ10/Mgt 10 Stuffler 57
GOOD NEWS! Go to Excel, Megastat, Probability, Discrete, Poisson.
Type in the mean, , 6, then OK. For a mean of 6,the Excel result is
0.00248 Slide 59 Econ10/Mgt 10 Stuffler 58 Poisson Distribution For
a 400 page book, what is the probability that there are five or
less typos? P(X5) = P(0) + P(1) + + P(5) Another alternative is to
refer to Table E-7 in the Appendices For x = 5, =6, and P(X 5) =
P(0) + + P(5) =.4456 There is about a 45% chance that there will be
5 or less typos Slide 60 Econ10/Mgt 10 Stuffler 59 Hypergeometric
Distribution If a sample is taken from a finite population without
replacement, the probability of X number of successes, follows the
hypergeometric distribution. NOTE: When you see without
replacement, this indicates that hypergeometric trials are
dependent. Slide 61 Econ10/Mgt 10 Stuffler 60 Hypergeometric
Distribution For both the Binomial Distribution and the
Hypergeometric Distribution, there are two possible outcomes for
the random discrete variable: Success or Failure Slide 62
Econ10/Mgt 10 Stuffler 61 Hypergeometric Distribution The
difference is that for Hypergeometric each trial is without
replacement, so the probability changes from one draw to the next
For Binomial, the probability of success and failure remain
constant since each trial or draw is independent Slide 63
Econ10/Mgt 10 Stuffler 62 Hypergeometric Distribution 1. Trials are
NOT independent 2. Finite population 3. Sample without replacement
4.Probability of success and failure changes from trial to trial
Slide 64 Econ10/Mgt 10 Stuffler 63 Hypergeometric Distribution r N
r = n r P(x) = x n x N N n 2 = r (N-r) n (N-n) N 2 (N-1) where N =
Total number in the population n = number in the sample, x = number
selected or drawn, r = number of successes in N Slide 65 Econ10/Mgt
10 Stuffler 64 Hypergeometric Distribution EXAMPLE: Assume that 3
stocks are chosen randomly from a list of 10 stocks and that of the
10 stocks 4 stocks pay dividends. The number (x) of the three
selected stocks that pay a dividend is a hypergeometric random
variable. Slide 66 Econ10/Mgt 10 Stuffler 65 Hypergeometric
Distribution N=10, n=3, r = 4 and x = number of the 3 stocks
selected that pay a dividend Calculate the mean, the variance and
the probability that none of the 3 stocks pay dividends? Slide 67
Econ10/Mgt 10 Stuffler 66 Hypergeometric Distribution 4 10 4 4! 6!
P(0) = 0 3-0 = 0!(4-0)! 3!(6-3)! = 1 10 10! 6 3 3!(10-3)! = (3)(4)
= 1.2 10 2 = 4(10-4)3(10-3) =.56, =.75 (10) 2 (10-1) Slide 68
Econ10/Mgt 10 Stuffler 67 Hypergeometric Distribution GOOD NEWS! Go
to ExcelMegastatProbability Discrete Probability Distributions
Hypergeometric, enter the required data Slide 69 Econ10/Mgt 10
Stuffler 68 Hypergeometric Distribution Slide 70 Econ10/Mgt 10
Stuffler 69 Discrete Probability WHICH DISTRIBUTION? Sony issued a
recall for their plasma screen TVs. They choose 25 TVs and 7 do not
operate. If the inspector chooses 3 at random from the 25, what is
the probability that 2 of the 3 do not operate? Slide 71 Econ10/Mgt
10 Stuffler 70 Discrete Probability WHICH DISTRIBUTION? When Wet
Water drills a well, their success rate is.55 and their profit is
$2,550. If they do not find water, they lose $1,500. Slide 72
Econ10/Mgt 10 Stuffler 71 Discrete Probability WHICH DISTRIBUTION?
The US unemployment rate is calculated from survey data of 60,000
households. What is the probability that no more than 5,000
respondents are unemployed? Slide 73 Econ10/Mgt 10 Stuffler 72
Discrete Probability WHICH DISTRIBUTION? US Transportation and
Safety Board reports the annual average number of plane crashes is
14.6. For a given month, what is the probability that at 2 crashes
will occur? That 3 crashes will occur? Slide 74 Econ10/Mgt 10
Stuffler 73 Discrete Probability WHICH DISTRIBUTION? The US
Statistical Abstract of the US tracks many data items such as, the
average number of automobiles per household. The average for 2009
is 2.5 autos. What is the probability that the 2010 average is at
least 3 autos? Slide 75 Econ10/Mgt 10 Stuffler 74 Discrete Random
Variables Two Types of Random Variables Discrete Probability
Distributions The Binomial Distribution The Poisson Distribution
The Hypergeometric Distribution Summary:
