Download 1 Chapter 5 Discrete Probability Distribution Learning objectives

Chapter 5
Discrete Probability Distribution
Slide 1
Learning objectives
1. Understand random variables and probability
distributions.
1.1. Distinguish discrete and continuous random
variables.
2. Able to compute Expected value and Variance of
discrete random variable.
3. Understand:
3.1. Discrete uniform distribution
3.2. Binomial distribution
3.3. Poisson distribution
Slide 2
1
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
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.
Slide 3
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
n
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)
n
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.
Slide 4
2
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
Random Variables
Question
Family
size
Type
Random Variable x
x = Number of dependents
reported on tax return
Discrete
Distance from x = Distance in miles from
home to store
home to the store site
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)
Slide 5
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
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.
Slide 6
3
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
Discrete Probability Distributions
The probability distribution for discrete random
variable 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
Σf(x) = 1
Slide 7
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
n
n
Discrete Probability Distributions
Using past data on TV sales, …
a tabular representation of the probability
distribution for TV sales was developed.
Units Sold
0
1
2
3
4
Number
of Days
80
50
40
10
20
200
x
0
1
2
3
4
f(x)
.40
.25
.20
.05
.10
1.00
80/200
Slide 8
4
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
n
Discrete Probability Distributions
Graphical Representation of Probability Distribution
Probability
.50
.40
.30
.20
.10
0
1
2
3
4
Values of Random Variable x (TV sales)
Slide 9
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
Expected Value and Variance
The expected value, or mean, of a random variable
is a measure of its central location.
E(x) = µ = Σxf(x)
The variance summarizes the variability in the
values of a random variable.
Var(x) = σ 2 = Σ(x - µ)2f(x)
The standard deviation, σ, is defined as the positive
square root of the variance.
Slide 10
5
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
n
Expected Value and Variance
Expected Value
x
0
1
2
3
4
f(x)
xf(x)
.40
.00
.25
.25
.20
.40
.05
.15
.10
.40
E(x) = 1.20
expected number of
TVs sold in a day
Slide 11
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
n
Expected Value and Variance
Variance and Standard Deviation
x
x-µ
0
1
2
3
4
-1.2
-0.2
0.8
1.8
2.8
(x - µ)2
f(x)
(x - µ)2f(x)
1.44
0.04
0.64
3.24
7.84
.40
.25
.20
.05
.10
.576
.010
.128
.162
.784
Variance of daily sales = σ 2 = 1.660
TVs
squared
Standard deviation of daily sales = 1.2884 TVs
Slide 12
6
In-class Exercise
n
Random variables
• # 2 (page 188)
• # 5 (page 188)
n
Expected value and variance
• #16 (page 196)
• #17 (page 197)
Slide 13
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
Discrete Uniform Probability Distribution
The discrete uniform probability distribution is the
simplest example of a discrete probability
distribution given by a formula.
The discrete uniform probability function is
f(x) = 1/n
the values of the
random variable
are equally likely
where:
n = the number of values the random
variable may assume
Slide 14
7
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
n
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.
Slide 15
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
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.
Slide 16
8
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
n
Binomial Distribution
Binomial Probability Function
n
f ( x ) =   px (1 − p )( n− x )
x
n!
=
p x (1 − p)( n −x )
x !( n − x )!
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
Slide 17
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
n
Binomial Distribution
Binomial Probability Function
f (x ) =
n!
px (1 − p)( n− x )
x !( n − x )!
n!
x !( n − x )!
Number of experimental
outcomes providing exactly
x successes in n trials
p x (1 − p )( n − x )
Probability of a particular
sequence of trial outcomes
with x successes in n trials
Slide 18
9
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
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.
Slide 19
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
n
Example: Evans Electronics
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
f ( x) =
f (1) =
n!
p x (1 − p ) (n − x )
x !( n − x )!
3!
(0.1)1 (0.9)2 = 3(.1)(.81) = .243
1!(3 − 1)!
Slide 20
10
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
n
Example: Evans Electronics
Tree Diagram
1st Worker
2nd Worker
Leaves (.1)
Leaves
(.1)
3rd Worker
L (.1)
x
3
Prob.
.0010
S (.9)
2
.0090
L (.1)
2
.0090
S (.9)
1
.0810
2
.0090
1
.0810
1
.0810
0
.7290
Stays (.9)
Leaves (.1)
Stays
(.9)
L (.1)
S (.9)
L (.1)
Stays (.9)
S (.9)
Slide 21
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
n
Example: Evans Electronics
Using Tables of Binomial Probabilities
p
n
x
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
3
0
1
2
3
.8574
.1354
.0071
.0001
.7290
.2430
.0270
.0010
.6141
.3251
.0574
.0034
.5120
.3840
.0960
.0080
.4219
.4219
.1406
.0156
.3430
.4410
.1890
.0270
.2746
.4436
.2389
.0429
.2160
.4320
.2880
.0640
.1664
.4084
.3341
.0911
.1250
.3750
.3750
.1250
Slide 22
11
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
n
Binomial Distribution
Expected Value
E(x) = µ = np
n
Variance
Var(x) = σ 2 = np(1 − p)
n
Standard Deviation
σ = np(1 − p )
Slide 23
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
n
Example: Evans Electronics
Expected Value
E(x) = µ = 3(.1) = .3 employees out of 3
n
Variance
Var(x) = σ 2 = 3(.1)(.9) = .27
n
Standard Deviation
σ = 3(.1)(.9) = .52 employees
Slide 24
12
In-class Exercise
n
n
#26 (page 207)
#37 (page 208)
Slide 25
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
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, . . . ).
Slide 26
13
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
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
Slide 27
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
n
Poisson Distribution
Two Properties of a Poisson Experiment
1. The probability of an occurrence is the same
for any two intervals of equal length.
2. The occurrence or nonoccurrence in any
interval is independent of the occurrence or
nonoccurrence in any other interval.
Slide 28
14
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
n
Poisson Distribution
Poisson Probability Function
µ x e−µ
f ( x) =
x!
where:
f(x) = probability of x occurrences in an interval
µ = mean number of occurrences in an interval
e = 2.71828
Slide 29
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
Example: Mercy Hospital
MERCY
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?
Slide 30
15
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
n
Example: Mercy Hospital
MERCY
Using the Poisson Probability Function
µ = 6/hour = 3/half-hour, x = 4
f (4) =
34 (2.71828)−3
= .1680
4!
Slide 31
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
n
x
0
1
2
3
4
5
6
7
8
Example: Mercy Hospital
MERCY
Using Poisson Probability Tables
2.1
.1225
.2572
.2700
.1890
.0992
.0417
.0146
.0044
.0011
2.2
.1108
.2438
.2681
.1966
.1082
.0476
.0174
.0055
.0015
2.3
.1003
.2306
.2652
.2033
.1169
.0538
.0206
.0068
.0019
2.4
.0907
.2177
.2613
.2090
.1254
.0602
.0241
.0083
.0025
µ
2.5
2.6
.0821 .0743
.2052 .1931
.2565 .2510
.2138 .2176
.1336 .1414
..0668 .0735
.0278 .0319
.0099 .0118
.0031 .0038
2.7
.0672
.1815
.2450
.2205
.1488
.0804
.0362
.0139
.0047
2.8
.0608
.1703
.2384
.2225
.1557
.0872
.0407
.0163
.0057
2.9
.0550
.1596
.2314
.2237
.1622
.0940
.0455
.0188
.0068
3.0
.0498
.1494
.2240
.2240
.1680
.1008
.0504
.0216
.0081
Slide 32
16
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
n
MERCY
Example: Mercy Hospital
Poisson Distribution of Arrivals
Poisson Probabilities
Probability
0.25
0.20
actually,
the sequence
continues:
11, 12, …
0.15
0.10
0.05
0.00
0
1
2
3
4
5
6
7
8
9
10
Number of Arrivals in 30 Minutes
Slide 33
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
Poisson Distribution
A property of the Poisson distribution is that
the mean and variance are equal.
µ=σ2
Slide 34
17
•L.O.
•Random variables
•EV and Variance
•Uniform discrete dist.
•Binomial dist.
•Poisson dist.
n
Example: Mercy Hospital
MERCY
Variance for Number of Arrivals
During 30-Minute Periods
µ=σ2=3
Slide 35
In-class Exercise
n
n
#38 (page 211)
#41 (page 212)
Slide 36
18
End of Chapter 5
Slide 37
19