Matakuliah
Tahun
: D0722 - Statistika dan Aplikasinya
: 2010
Peubah Acak
Pertemuan 3
Learning Outcomes
•
Pada akhir pertemuan ini, diharapkan
mahasiswa akan mampu :
1. menghasilkan sebaran variabel acak
diskrit, nilai harapan , sebaran binomial
dan poisson
2. memperhitungkan sebaran kontinyu,
sebaran seragam dan eksponensial
3
COMPLETE
BUSINESS STATISTICS
1-4
5th edi tion
Random Variables







Using Statistics
Expected Values of Discrete Random
Variables
The Binomial Distribution
Other Discrete Probability Distributions
Continuous Random Variables
Using the Computer
Summary and Review of Terms
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
5th edi tion
1-5
BUSINESS STATISTICS
Using Statistics
Consider the different possible orderings of boy (B) and girl (G) in
four sequential births. There are 2*2*2*2=24 = 16 possibilities, so
the sample space is:
BBBB
BBBG
BBGB
BBGG
BGBB
BGBG
BGGB
BGGG
GBBB
GBBG
GBGB
GBGG
GGBB
GGBG
GGGB
GGGG
If girl and boy are each equally likely [P(G)=P(B) = 1/2], and the
gender of each child is independent of that of the previous child,
then the probability of each of these 16 possibilities is:
(1/2)(1/2)(1/2)(1/2) = 1/16.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
5th edi tion
1-6
BUSINESS STATISTICS
Random Variables
Now count the number of girls in each set of four sequential births:
BBBB
BBBG
BBGB
BBGG
(0)
(1)
(1)
(2)
BGBB
BGBG
BGGB
BGGG
(1)
(2)
(2)
(3)
GBBB
GBBG
GBGB
GBGG
(1)
(2)
(2)
(3)
GGBB
GGBG
GGGB
GGGG
(2)
(3)
(3)
(4)
Notice that:
•
each possible outcome is assigned a single numeric value,
•
all outcomes are assigned a numeric value, and
•
the value assigned varies over the outcomes.
The count of the number of girls is a random variable:
A random variable, X, is a function that assigns a single, but variable, value to
each element of a sample space.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
5th edi tion
1-7
BUSINESS STATISTICS
Random Variables (Continued)
0
BBBB
BGBB
GBBB
BBBG
BBGB
GGBB
GBBG
BGBG
BGGB
GBGB
BBGG
BGGG
GBGG
GGGB
GGBG
GGGG
1
X
2
3
4
Sample Space
McGraw-Hill/Irwin
Aczel/Sounderpandian
Points on the
Real Line
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
5th edi tion
1-8
BUSINESS STATISTICS
Random Variables (Continued)
Since the random variable X = 3 when any of the four outcomes BGGG, GBGG,
GGBG, or GGGB occurs,
P(X = 3) = P(BGGG) + P(GBGG) + P(GGBG) + P(GGGB) = 4/16
The probability distribution of a random variable is a table that lists the
possible values of the random variables and their associated probabilities.
P(x)
1/16
4/16
6/16
4/16
1/16
16/16=1
P ro b a b ility D is trib u tio n o f th e N u m b e r o f G irls in F o u r B irth s
0 .4
0 .3 7 5 0
0 .3
0 .2 5 0 0
P(x)
x
0
1
2
3
4
0 .2 5 0 0
0 .2
0 .1
0 .0 6 2 5
0
0 .0 6 2 5
1
2
3
4
N u m b e r o f g i rls , x
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
1-9
5th edi tion
Discrete and Continuous Random
Variables
A discrete random variable:

has a countable number of possible values

has discrete jumps (or gaps) between successive values

has measurable probability associated with individual values

counts
A continuous random variable:

has an uncountably infinite number of possible values

moves continuously from value to value

has no measurable probability associated with each value
 measures (e.g.: height, weight, speed, value, duration, length)
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
1-10
BUSINESS STATISTICS
5th edi tion
Rules of Discrete Probability
Distributions
The probability distribution of a discrete random
variable X must satisfy the following two conditions.
1. P(x)  0 for all values of x.
2.
 P(x)  1
all x
Corollary: 0  P( X )  1
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
5th edi tion
1-11
Cumulative Distribution Function
The cumulative distribution function, F(x), of a discrete
random variable X is:
F(x)  P( X  x) 
 P(i)
all i  x
McGraw-Hill/Irwin
P(x)
0.1
0.2
0.3
0.2
0.1
0.1
1
F(x)
0.1
0.3
0.6
0.8
0.9
1.0
Cumulative Probability Distribution of the Number of Switches
1 .0
0 .9
0 .8
0 .7
F(x)
x
0
1
2
3
4
5
0 .6
0 .5
0 .4
0 .3
0 .2
0 .1
0 .0
0
1
2
3
4
5
x
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
5th edi tion
1-12
BUSINESS STATISTICS
Expected Values of Discrete Random
Variables
The mean of a probability distribution is a
measure of its centrality or location, as is the
mean or average of a frequency distribution. It is
a weighted average, with the values of the
random variable weighted by their probabilities.
0
1
2
3
4
5
2.3
The mean is also known as the expected value (or expectation) of a random
variable, because it is the value that is expected to occur, on average.
The expected value of a discrete random
variable X is equal to the sum of each
value of the random variable multiplied by
its probability.
  E ( X )   xP ( x )
all x
McGraw-Hill/Irwin
Aczel/Sounderpandian
x
0
1
2
3
4
5
P(x)
0.1
0.2
0.3
0.2
0.1
0.1
1.0
xP(x)
0.0
0.2
0.6
0.6
0.4
0.5
2.3 = E(X) = 
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
5th edi tion
1-13
Variance and Standard Deviation of a
Random Variable
The variance of a random variable is the expected
squared deviation from the mean:

2
 V ( X )  E [( X   ) 2 ] 

(x   ) 2 P(x)
a ll x

 

 E ( X 2 )  [ E ( X )] 2    x 2 P ( x )     xP ( x ) 
 a ll x
  a ll x

2
The standard deviation of a random variable is the
square root of its variance:   SD( X )  V ( X )
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
1-14
5th edi tion
Bernoulli Random Variable
• If an experiment consists of a single trial and the outcome of the
trial can only be either a success* or a failure, then the trial is
called a Bernoulli trial.
• The number of success X in one Bernoulli trial, which can be 1 or
0, is a Bernoulli random variable.
• Note: If p is the probability of success in a Bernoulli experiment,
the E(X) = p and V(X) = p(1 – p).
* The
terms success and failure are simply statistical terms, and do not have positive or
negative implications. In a production setting, finding a defective product may be termed
a “success,” although it is not a positive result.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
1-15
5th edi tion
The Binomial Random Variable
Consider a Bernoulli Process in which we have a sequence of n identical
trials satisfying the following conditions:
1. Each trial has two possible outcomes, called success *and failure.
The two outcomes are mutually exclusive and exhaustive.
2. The probability of success, denoted by p, remains constant from trial
to trial. The probability of failure is denoted by q, where q = 1-p.
3. The n trials are independent. That is, the outcome of any trial does
not affect the outcomes of the other trials.
A random variable, X, that counts the number of successes in n Bernoulli
trials, where p is the probability of success* in any given trial, is said to
follow the binomial probability distribution with parameters n (number
of trials) and p (probability of success). We call X the binomial random
variable.
* The terms success and failure are simply statistical terms, and do not have positive or negative implications. In a
production setting, finding a defective product may be termed a “success,” although it is not a positive result.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
5th edi tion
1-16
BUSINESS STATISTICS
The Binomial Probability Distribution
The binomial probability distribution:
n!
 n x ( n  x )
P( x)    p q

p x q ( n x)
x!( n  x)!
 x
where :
p is the probability of success in a single trial,
q = 1-p,
n is the number of trials, and
x is the number of successes.
N u m b er o f
su ccesses, x
0
1
2
3

n
P ro b ab ility P (x )
n!
p 0 q (n 0)
0 !( n  0 ) !
n!
p 1 q ( n  1)
1 !( n  1 ) !
n!
p 2 q (n 2)
2 !( n  2 ) !
n!
p 3 q (n 3)
3 !( n  3 ) !

n!
p n q (n n)
n !( n  n ) !
1 .0 0
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
5th edi tion
1-17
BUSINESS STATISTICS
Mean, Variance, and Standard Deviation
of the Binomial Distribution
Mean of a binomial distribution:
  E ( X )  np
For example, if H counts the number of
heads in five tosses of a fair coin :
Variance of a binomial distribution:
  E ( H )  (5)(.5)  2.5
H
  V ( X )  npq
2
  V ( H )  (5)(.5)(.5)  1.25
2
H
Standard deviation of a binomial distribution:
 = SD(X) = npq
McGraw-Hill/Irwin
Aczel/Sounderpandian
  SD( H )  1.25  1.118
H
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
1-18
5th edi tion
The Poisson Distribution
The Poisson probability distribution is useful for determining the probability of a
number of occurrences over a given period of time or within a given area or
volume. That is, the Poisson random variable counts occurrences over a
continuous interval of time or space. It can also be used to calculate approximate
binomial probabilities when the probability of success is small (p0.05) and the
number of trials is large (n20).
Poisson D istribution:
 xe 
P( x) 
for x = 1,2,3,...
x!
where  is the mean of the distribution (which also happens to be the variance) and
e is the base of natural logarithms (e=2.71828...).
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
1-19
5th edi tion
The Poisson Distribution (continued)
• Poisson assumptions:
The probability that an event will occur in a short
interval of time or space is proportional to the size of
the interval.
In a very small interval, the probability that two events
will occur is close to zero.
The probability that any number of events will occur in
a given interval is independent of where the interval
begins.
The probability of any number of events occurring over
a given interval is independent of the number of events
that occurred prior to the interval.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
1-20
5th edi tion
The Poisson Distribution - Example
Example 3-5:
Telephone manufacturers now offer 1000
different choices for a telephone (as combinations
of color, type, options, portability, etc.). A
company is opening a large regional office, and
each of its 200 managers is allowed to order his
or her own choice of a telephone. Assuming
independence of choices and that each of the
1000 choices is equally likely, what is the
probability that a particular choice will be made
by none, one, two, or three of the managers?

.2 0 e .2
P ( 0) 
= 0.8187
0
!

.21 e .2
P (1) 
= 0.1637
12 !  .2
.2 e
P (2) 
= 0.0164
2
!

.2 3 e .2
P ( 3) 
= 0.0011
3!
n = 200  = np = (200)(0.001) = 0.2
p = 1/1000 = 0.001
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
5th edi tion
1-21
BUSINESS STATISTICS
Discrete and Continuous Random
Variables - Revisited
• A continuous random variable:
–
–
–
–
–
counts occurrences
has a countable number of possible
values
has discrete jumps between
successive values
has measurable probability
associated with individual values
probability is height
For example:
Binomial
n=3 p=.5
Binomial: n=3 p=.5
0.4
P(x)
0.125
0.375
0.375
0.125
1.000
McGraw-Hill/Irwin
P(x)
0.3
x
0
1
2
3
0.2
0.1
0.0
0
1
2
3
C1
Aczel/Sounderpandian
–
–
–
–
–
measures (e.g.: height, weight,
speed, value, duration, length)
has an uncountably infinite number
of possible values
moves continuously from value to
value
has no measurable probability
associated with individual values
probability is area
For example:
In this case,
the shaded
area epresents
the probability
that the task
takes between
2 and 3
minutes.
Minutes to Complete Task
0.3
0.2
P(x)
• A discrete random variable:
0.1
0.0
1
2
3
4
5
6
Minutes
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
5th edi tion
1-22
BUSINESS STATISTICS
From a Discrete to a Continuous
Distribution
The time it takes to complete a task can be subdivided into:
Half-Minute Intervals
Eighth-Minute Intervals
Quarter-Minute Intervals
Minutes to Complete Task: Fourths of a Minute
Minutes to Complete Task: By Half-Minutes
Minutes to Complete Task: Eighths of aMinute
0.15
P(x)
P(x)
P(x)
0.10
0.05
0.00
0.0
. 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5
0
Minutes
1
2
3
4
5
Minutes
6
7
0
1
2
3
4
5
6
7
Minutes
Or even infinitesimally small intervals:
f(z)
Minutes to Complete Task: Probability Density Function
0
1
2
3
4
5
6
When a continuous random variable has been subdivided into
infinitesimally small intervals, a measurable probability can
only be associated with an interval of values, and the
probability is given by the area beneath the probability density
function corresponding to that interval. In this example, the
shaded area represents P(2  X  ).
7
Minutes
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
1-23
5th edi tion
Continuous Random Variables
A continuous random variable is a random variable that can take on any value in an
interval of numbers.
The probabilities associated with a continuous random variable X are determined by the
probability density function of the random variable. The function, denoted f(x), has the
following properties.
1.
2.
3.
f(x)  0 for all x.
The probability that X will be between two numbers a and b is equal to the area
under f(x) between a and b.
The total area under the curve of f(x) is equal to 1.00.
The cumulative distribution function of a continuous random variable:
F(x) = P(X  x) =Area under f(x) between the smallest possible value of X (often -) and
the point x.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
5th edi tion
1-24
BUSINESS STATISTICS
Probability Density Function and
Cumulative Distribution Function
F(x)
1
F(b)
}
F(a)
P(a  X  b)=F(b) - F(a)
0
a
x
b
f(x)
P(a  X  b) = Area under
f(x) between a and b
= F(b) - F(a)
0
McGraw-Hill/Irwin
a
b
Aczel/Sounderpandian
x
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
1-25
BUSINESS STATISTICS
5th edi tion
3-10 Uniform Distribution
The uniform [a,b] density:
1/(a – b) for a  X  b
f(x)=
{
0 otherwise
E(X) = (a + b)/2; V(X) = (b – a)2/12
Uniform [a, b] Distribution
f(x)
The entire area under f(x) = 1/(b – a) * (b – a) = 1.00
The area under f(x) from a1 to b1 = P(a1X b1)
= (b1 – a1)/(b –
a)
a a1
McGraw-Hill/Irwin
b1
b
x
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
5th edi tion
1-26
BUSINESS STATISTICS
Exponential Distribution
E xp onential D is tributio n: = 2
The exponential random variable measures the
time between two occurrences that have a
Poisson distribution.
2
Exponential distribution:
f(x)
The density function is:
f (x)  ex for x  0,   0
1
1
The mean and standard deviation are both equal to .

0
The cumulative distribution function is:
0
F(x)  1  ex for x  0.
McGraw-Hill/Irwin
1
2
3
Time
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
RINGKASAN
Variabel Acak diskrit:
- Nilai harapan variabel acak diskrit
- Variabel acak Binomial
- Variabel acak Poisson
Variabel acak kontinyu:
- Sebaran seragam
- Sebaran eksponensial
27