Transcript Poisson distribution

Al-Imam Mohammad Ibn Saud University
CS433
Modeling and Simulation
Lecture 04
Statistical Models
http://10.2.230.10:4040/akoubaa/cs433/
27 Oct 2008
Dr. Anis Koubâa
1
Goals for Today
 Understand the difference between
discrete and continuous random
variables
 Review of the most common statistical
models
 Understand how to determine the
empirical distribution from a statistical
sample.
2
2
Topics
 Discrete Random Variable
 Continuous Random Variable
 Discrete Probability Distributions
 Binomial Distribution
 Bernoulli Distribution
 Discrete Poisson Distribution
 Continuous Probability Distribution
 Uniform
 Exponential
 Normal
 Weibull
 Lognormal
 Empirical Distributions
3
3
Discrete and Continuous Random Variables
4
Discrete Random Variables


X is a discrete random variable if the number of possible values
of X (the sample space) is finite.
Example: Consider jobs arriving at a job shop.



Let X be the number of jobs arriving each week at a job shop.
Rx = possible values of X (range space of X) = {0,1,2,…}
p(xi) = probability the random variable is xi = p(X = xi)

The collection of pairs [xi, p(xi)], i = 1,2,…, is called the probability
distribution of X,
p(xi) is called the probability mass function (PMF) of X.

Characteristics of the PMF: p(xi), i = 1,2, … must satisfy:

1. p (x i )  0, for all i
2.


i 1
p (x i )  1
5
Continuous Random Variables


X is a continuous random variable if its range space Rx is an interval or a
collection of intervals.
The probability that X lies in the interval [a,b] is given by:
b
P(a  X  b)   f ( x)dx
a
Where f(x) is the probability density function (PDF).

Characteristics of the PDF: f(x) must satisfies:
1. f ( x)  0 , for all x in R X
2.
 f ( x)dx  1
RX
3. f ( x)  0, if x is not in RX

Properties
x0
1. P( X  x0 )  0, because  f ( x)dx  0
x0
2. P(a  X  b)  P(a  X  b)  P(a  X  b)  P(a  X  b)
6
Discrete versus Continuous
Random Variables
Discrete Random Variable
Continuous Random Variable
Finite Sample Space
e.g. {0, 1, 2, 3}
Probability Mass Function (PMF)
p x i   P X  x i 
1. p (x i )  0, for all i
2.  i 1 p (x i )  1

Infinite Sample Space
e.g. [0,1], [2.1, 5.3]
Probability Density Function (PDF)
f x 
1. f ( x)  0 , for all x in R X
2.
 f ( x)dx  1
RX
3. f ( x)  0, if x is not in RX
Cumulative Distribution Function (CDF) p  X  x 
p  X  x   x
i
x
p (x i )
p  X  x    f t  dt  0
x

p a  X  b    f  x  dx
b
a
7
Five Minutes Break
You are free to discuss with your classmates about
the previous slides, or to refresh a bit, or to ask
questions.
Administrative issues
• Groups Formation
• Choose a “class coordinator”
8
Expectation

The expected value (the mean) of X is denoted by E(X)

If X is discrete
E ( x)   xi p( xi )
all i



If X is continuous

A measure of the central tendency

The variance of X is denoted by V(X) or var(X) or s2




E ( x)   xf ( x)dx
2


V
X

E
X

E
X




Definition:


2
Also,
V  X   E  X 2    E  X 
A measure of the spread or variation of the possible values of X around the
mean
The standard deviation of X is denoted by s


Definition: square root of V(X)
Expressed in the same units as the mean
9
Example: Continuous Random Variables
Example: modeling the lifetime of a device

Time is a continuous random variable

Random Time is typically modeled as exponential distribution

We assume that with average lifetime of a device is 2 years
 1  x / 2
e , x0
f ( x)   2
0,
otherwise

Probability that the device’s life is between 2 and 3 years is:
1 3 x / 2
P(2  x  3)   e dx  0.14
2 2
10
Example: Continuous Random Variables

Cumulative Distribution Function: A device has the CDF:
1 x t / 2
F ( x)   e dt  1  e  x / 2
2 0

The probability that the device lasts for less than 2 years:
P(0  X  2)  F (2)  F (0)  F (2)  1  e1  0.632

The probability that it lasts between 2 and 3 years:
P(2  X  3)  F (3)  F (2)  (1  e (3 / 2) )  (1  e 1 )  0.145
11
Example: Continuous Random Variables
Expected Value and Variance


Example: The mean of life of the previous device is:


1  x /2
x /2
E (X )   xe dx  xe
  e  x / 2dx  2
0
2 0
0
To compute variance of X, we first compute E(X2):


1
x / 2
2 x / 2
2
E ( X )   x e dx   x e
  e  x / 2 dx  8
0
2 0
0
2


Hence, the variance and standard deviation of the device’s life are:
V ( X )  8  22  4
s  V (X )  2
12
Discrete Probability Distributions
Bernoulli Trials
 Binomial Distribution
 Geometric Distribution
 Poisson Distribution
 Poisson Process

13
Discrete Distributions


Discrete random variables are used to describe
random phenomena in which only integer values can
occur.
In this section, we will learn about:
 Bernoulli trials and Bernoulli distribution
 Binomial distribution
 Geometric and negative binomial distribution
 Poisson distribution
14
14
Modeling of Random Events with Two-States
Bernoulli Trials
 Binomial Distribution

15
Bernoulli Trials


In the theory of probability and statistics, a Bernoulli
trial is an experiment whose outcome is random and can
be either of two possible outcomes, "success" and
"failure".
In practice it refers to a single experiment, which can
have one of two possible outcomes. These events can be
phrased into “yes” or “no” questions:
Did the coin land heads?
 Was the newborn child or a girl?
 Were a person's eyes green?

16
Bernoulli Distribution

Consider an experiment consisting of n trials, each can be a
success or a failure.



Let Xj = 1 if the jth experiment is a success
and Xj = 0 if the jth experiment is a failure
The Bernoulli distribution (one trial):
x j  1, j  1, 2,..., n
p,

PMF: p j (x j )  p (x j )  1  p  q , x j  0 , j  1, 2 ,..., n
0,
otherwise

Expected Value: E  X j   p

Variance :V X j   s 2  p  1  p 
Bernoulli process

It is the n Bernoulli trials where trials are independent:
p  X 1, X 1,..., X n ,   p  X 1   p  X 2   ... p  X n 
17
Binomial Distribution

A binomial random variable is the number of successes in a
series of n trials.



Example: the number of 'heads' occurring when a coin is tossed 50 times.
A discrete random variable X is said to follow a Binomial
distribution with parameters n and p, written X ~ Bi(n,p) or
X ~ B(n,p) if it has the probability distribution:
n  k
n k
P  X  k     p  1  p 
k 
where



Where
x = 0, 1, 2, ......., n
n = 1, 2, 3, .......
p = success probability; 0 < p < 1
n 
n!
 
 k  k !  n  k !
Expected Value: E X   n  p
Variance :V  X   s 2  n  p  1  p 
18
Binomial Distribution

The trials must meet the following
requirements:




the total number of trials is fixed in advance;
there are just two outcomes of each trial;
success and failure;
the outcomes of all the trials are statistically
independent;
all the trials have the same probability of
success.
19
19
Binomial Distribution

The number of successes in n Bernoulli trials, X, has a binomial
distribution.
 n  k n  k
  p q
p ( X  k )   k 
0,

The number of
outcomes having the
required number of
successes and
failures

, k  0,1,2,..., n
otherwise
Probability that
there are
x successes and
(n-x) failures
The formula can be understood as follows: we want k successes
(pk) and n − k failures (1 − p)n − k. However, the k successes can
occur anywhere among the n trials, and there are C(n, k)
different ways of distributing k successes in a sequence of n trials.
20
End of Part 01
Administrative issues
• Groups Formation
• Choose a “class coordinator”
21
Modeling of Discrete Random Time

Geometric Distribution
22
Geometric Distribution

Geometric Distribution represents the number X of Bernoulli trials to
achieve the FIRST SUCCESS.

It is used to represent random time until a first transition occurs
q
PMF: p (X  k )  
0,
k 1
p , k  0,1, 2,..., n
PMF
otherwise
CDF: F  X   p  X  k   1  1  p 
Expected Value : E  X  
Variance :V  X   s 
2
k
1
p
k
q
p
2

1 p
p2
23

Negative Binomial Distribution
24
Negative Binomial Distribution



The negative binomial distribution is a discrete probability distribution that can be
used to describe the distribution arising from an experiment consisting of a
sequence of independent trials, subject to several constraints.
Firstly each trial results in success or failure, the probability of success for each
trial, p, is constant across the experiment and finally the experiment continues until
a fixed number of successes have been achieved.
Negative Binomial Distribution


The number of Bernoulli trials, X, until the kth success
If X is a negative binomial distribution with parameters p and r, then:
 k  r  1 r
k

p

1

p

 , k  1, 2,3...


PMF  r , p  : p  X  k    k

0,
otherwise

1 p
Expected Value : E  X   r
p
Variance :V  X   s 
2
r  1  p 
p2
25
Modeling of Random Number of
Arrivals/Events
Poisson Distribution
 Poisson Process

26
Poisson Distribution



the Poisson distribution is a discrete probability distribution that
expresses the probability of a number of events occurring in a
fixed period of time if these events occur with a known average
rate and independently of the time since the last event.
Poisson random variable represents the count of the number of
events that occur in a certain time interval or spatial area.
Example:
 The number of cars passing a fixed point in a 5 minute interval,
 The number of calls received by a switchboard during a given
period of time.
 The number of message coming to a router in a given period of
time
27
Discrete Poisson Distribution

A discrete random variable X is said to follow a Poisson
distribution with parameter l, written X ~ Po(l), if it has
probability distribution
PMF: P  X  k  


l
k
k!
exp  l 
The PMF represents the probability that there are k arrivals
in a certain period of time.
where
X = 0, 1, 2, ..., n
 l > 0 is called the arrival rate.

28
Discrete Poisson Distribution


Poisson distribution describes many random processes quite well
and is mathematically quite simple.
The Poisson distribution with the parameter l is characterized by:
lk
exp  l  for k  0,1, 2, ....

PMF: p  k   P  X  k    k !
0,
otherwise

CDF: F  k   p  X  k  
k
PMF
li
 i !  exp  l 
i 0
Expected value: E  X   l
Variance: V  X   l
CDF
29
Discrete Poisson Distribution

The following requirements must be met in the Poisson Distribution:

the length of the observation period is fixed in advance;

the events occur at a constant average rate;

the number of events occurring in disjoint intervals are
statistically independent.
30
Example: Poisson Distribution

The number of cars that enter the parking follows a Poisson
distribution with a mean rate equal to l = 20 cars/hour

or

The probability of having exactly 15 cars entering the parking in one hour:
2015
p 15  P  X  15 
 exp  20   0.051649
15!
p 15  F 15  F 14  0.156513  0.104864  0.051649
The probability of having more than 3 cars entering the parking in one hour:
p  X  3  1  p  X  3  1  F  3
 1   p  0   p 1  p  2   p  3 
 0.9999967
USE EXCEL/MATLAB
FOR COMPUTATIONS
31
Example: Poisson Distribution
Probability Mass Function
Poisson (l = 20 cars/hour)
20k
p X  k  
 exp  20 
k!
Cumulative Distribution Function
Poisson (l = 20 cars/hour)
k
20i
F k   p X  k  
 exp  20 
i!
i 0

32
Five Minutes Break
You are free to discuss with your classmates about
the previous slides, or to refresh a bit, or to ask
questions.
Administrative issues
• Groups Formation
33
Modeling of Random Number of
Arrivals/Events
Poisson Distribution
 Poisson Process

34
Poisson Process




Wikipedia: A Poisson process, named after the French
mathematician Siméon-Denis Poisson (1781 – 1840), is the
stochastic process in which events (e.g. arrivals) occur continuously
and independently of one another.
Formal Definition: The Poisson Process is a counting function {N(t),
t≥0} where N(t) is the number of events that have occurred up to
time t , i.e. in the interval [0, t].
Fact: The number of events between time a and time b is given as
N(b) − N(a) and has a Poisson distribution.
The Poisson process is a continuous-time process: Time is continuous

Its discrete-time counterpart is the Bernoulli process

Bernoulli process is a discrete-time stochastic process consisting of a sequence of
independent random variables taking values over two symbols.
35
Examples of using Poisson Process





The number of web page requests arriving at a server may be
characterized by a Poisson process except for unusual circumstances
such as coordinated denial of service attacks.
The number of telephone calls arriving at a switchboard, or at an
automatic phone-switching system, may be characterized by a
Poisson process.
The number of raindrops falling over a wide spatial area may be
characterized by a spatial Poisson process.
The arrival of "customers" is commonly modelled as a Poisson process
in the study of simple queueing systems.
The execution of trades on a stock exchange, as viewed on a tick by
tick basis, is a Poisson process.
36
(Homogenous) Poisson Process


The homogeneous Poisson process is characterized by a CONSTANT
rate parameter λ, also known as intensity, such that the number of
events in time interval t , t  t  follows a Poisson distribution with
associated parameter l t .
Formally, A counting process N t  ,t  0 is a (homogenous)
Poisson process with mean rate l if:


for t  0 and n  0,1, 2,...
(l  t ) n
PMF: p  N t  t   N t   n   p  N (t )  n  
exp  l t 
n!


N t  t   N t  describes the number of events in time interval t , t  t 
The mean and the variance are equal E  N t  V  N t   l t
37
(Homogenous) Poisson Process

Properties of Poisson process
Arrivals occur one at a time (not simultaneous)
 N t  ,t  0 has stationary increments, which means
N t   N  s   N t  s 
The number of arrivals in time s to t is also Poissondistributed with mean l  t  s 
 N t  ,t  0 has independent increments

38
CDF of Exponential distribution
Inter-Arrival Times of a Poisson Process
39



Inter-arrival time: time between two consecutive arrivals
 The inter-arrival times of a Poisson process are random.
What is its distribution?
Consider the inter-arrival times of a Poisson process (A1, A2, …), where Ai is the
elapsed time between arrival i and arrival i+1
The first arrival occurs after time t MEANS that there are no
arrivals in the interval [0,t], As a consequence:
p  A1  t   p  N t   0   exp  l  t 
p  A1  t   1  p  A1  t   1  exp  l  t 
The Inter-arrival times of a Poisson process are
exponentially distributed and independent with mean 1/l
39
Splitting and Pooling

Splitting


A Poisson process can be split into two Poisson processes: The first with a
probability p and the second with probability 1-p.
N t   N 1 t   N 2 t  where N 1 t  and N 2 t  are both Poisson processes with
rates l  p and l  1  p 
N(t) ~ Poi(l)
lp
l
l(1-p)

N1(t) ~ Poi[lp]
N2(t) ~ Poi[l(1-p)]
Pooling


The summation of two Poisson processes is a Poisson process
N 1 t   N 2 t   N t  , where N t  is a Poisson processes with rates l1  l2
N1(t) ~ Poi[l1]
N2(t) ~ Poi[l2]
l1
l 1  l2
l2
N(t) ~ Poi(l1  l2)
40
Modeling of Random Number of
Arrivals/Events
Poisson Distribution
 Non Homogenous Poisson
Process

41
Non Homogenous (Non-stationary) Poisson Process (NSPP)

The non homogeneous Poisson process is characterized
by a VARIABLE rate parameter λ(t), the arrival rate at
time t. In general, the rate parameter may change over
time.
l1


l2
l3
The stationary increments, property is not satisfied
s ,t : N t   N s   N t  s 
The expected number of events (e.g. arrival) between
time s and time t is
t
ls ,t   λ(u)  du
s
42
Example: Non-stationary Poisson Process (NSPP)

The number of cars that cross the intersection of King Fahd Road and Al-Ourouba
Road is distributed according to a non homogenous Poisson process with a mean
l(t) defined as follows:
80 cars/mn if 8 am  t  9am

60 cars/mn if 9am  t  11pm
l t   
50 car/mn if 11am  t  15 pm
70 car/mn if 15 pm  t  17 pm


Let us consider the time 8 am as t=0.

Q1. Compute the average arrival number of cars at 11H30?


Q2. Determine the equation that gives the probability of having only 10000 car
arrivals between 12 pm and 16 pm.
Q3. What is the distribution and the average (in seconds) of the inter-arrival time
of two cars between 8 am and 9 am?
43
Example: Non-stationary Poisson Process (NSPP)

Q1. Compute the average arrival number of cars at 11H30?
l8:00,11:30  
11:30
8:00

9:00
8:00
λ(u)  du
λ(u)  du  
11:00
9:00
λ(u)  du  
11:30
11:00
λ(u)  du
  80cars/mn  60mn    60cars/mn 120mn    50cars/mn  30mn   13500 cars

Q2. Determine the equation that gives the probability of having only 10000 car
arrivals between 12 pm and 16 pm.
We know that the number of cars between 12 pm and 16 pm, i.e.  N 16   N 12  
follows a Poisson distribution. During 12 pm and 16pm, the average number of
cars is l12:0016:00  180  50  60  70  13200 cars
Thus,

p  N 16   N 12 
13200 
 10000 
10000

 exp  13200 
10000!
Q3. What is the distribution and the average (in seconds) of the inter-arrival time
44
of two cars between 8 am and 9 am? (Homework)
Two Minutes Break
You are free to discuss with your classmates about
the previous slides, or to refresh a bit, or to ask
questions.
Administrative issues
• Groups Formation
45
Continuous Probability Distributions
Uniform Distribution
 Exponential Distribution
 Normal (Gaussian) Distribution
 Weibull Distribution
Lognormal Distribution

46
Continuous Distributions


Continuous random variables can be used to
describe random phenomena in which the variable
can take on any value in some interval.
In this section, the distributions studied are:
 Uniform
 Exponential
 Normal
 Weibull
 Lognormal
47
Uniform Distribution
48
Continuous Uniform Distribution


The continuous uniform distribution is a family of probability
distributions such that for each member of the family, all intervals of
the same length on the distribution's support are equally probable
A random variable X is uniformly distributed on the interval [a,b],
U(a,b), if its PDF and CDF are:
 1
, a  x b

PDF: f ( x )  b  a
0,
otherwise
a b
Expected value: E  X  
2
x a
0,
x a

CDF: F (x )  
, ax
b  a
x b
1,
Variance: V  X  
b
 a  b 2
12
49
Uniform Distribution
PDF

Properties
 p  x 1  X  x 2 is proportional to the
length of the interval
F X 2   F X 1  

X 2 X1
b a
Special case: a standard uniform
distribution U(0,1).

CDF
Very useful for random number
generators in simulators
50
Exponential Distribution

Modeling Random Time
51
Exponential Distribution


The exponential distribution describes the times between events
in a Poisson process, in which events occur continuously and
independently at a constant average rate.
A random variable X is exponentially distributed with parameter
m1/l > 0 if its PDF and CDF are:
1
 x

exp
x 0
l  exp  l  x  ,
x 0

  ,
PDF: f (x ) 
f (x ) 

0,
otherwise
m
0,

 m
otherwise
x 0
0,
CDF: F (x )   x  lt
lx
 0 le dt  1  e , x  0
0,

F (x )  
 x
1

exp
  ,

 m

Expected value: E  X  
Variance: V  X  
1
l
m
1
l2
x 0
x 0
 m2
52
Exponential Distribution
µ=20
1
 x 

exp
x 0

  ,
f (x )   20
20


0,
otherwise

µ=20
0,

F (x )  
 x 
1

exp
  ,

 20 

x 0
x 0
53
Exponential Distribution



The memoryless property: In probability theory, memoryless is a
property of certain probability distributions: the exponential
distributions and the geometric distributions, wherein any derived
probability from a set of random samples is distinct and has no
information (i.e. "memory") of earlier samples.
Formally, the memoryless property is:
For all s and t greater or equal to 0:
p X  s  t | X  s   p X  t 
This means that the future event do not depend on the past event,
but only on the present event

The fact that Pr(X > 40 | X > 30) = Pr(X > 10) does not mean that the
events X > 40 and X > 30 are independent; i.e. it does not mean that
Pr(X > 40 | X > 30) = Pr(X > 40).
54
Exponential Distribution


The memoryless property: can be read as “the probability that
you will wait more than s+t minutes given that you have already
been waiting t minutes is equal to the probability that you will wait
s minutes.”
In other words “The probability that you will wait s more minutes
given that you have already been waiting t minutes is the same as
the probability that you had wait for more than s minutes from the
beginning.”
p X  s  t | X  s   p X  t 

The fact that Pr(X > 40 | X > 30) = Pr(X > 10) does not mean that the events
X > 40 and X > 30 are independent; i.e. it does not mean that
Pr(X > 40 | X > 30) = Pr(X > 40).
55
Example: Exponential Distribution

The time needed to repair the engine of a car is exponentially
distributed with a mean time equal to 3 hours.

The probability that the car spends more than 3 hours in reparation

 3 
p  X  3  1  p  X  3  1  F  3  1  1  exp      0.368
 3 


The probability that the car repair time lasts between 2 to 3 hours is:
p   X  3  F 3  F  2  0.145

The probability that the repair time lasts for another hour given it has been
operating for 2.5 hours:
Using the memoryless property of the exponential distribution, we have:
 1
p  X   2.5  1 | X  2.5   p  X  1  1  p  X  1  exp     0.717
 3
56
Normal (Gaussian) Distribution
57
Normal Distribution




The Normal distribution, also called the Gaussian distribution, is
an important family of continuous probability distributions,
applicable in many fields.
Each member of the family may be defined by two parameters,
location and scale: the mean ("average", μ) and variance
(standard deviation squared, σ2) respectively.
The importance of the normal distribution as a model of
quantitative phenomena in the natural and behavioral sciences is
due in part to the Central Limit Theorem.
It is usually used to model system error (e.g. channel error), the
distribution of natural phenomena, height, weight, etc.
58
Normal or Gaussian Distribution

A continuous random variable X, taking all real values in the
range (-∞,+∞) is said to follow a Normal distribution with
parameters µ and σ if it has the following PDF and CDF:
 1  x  m 2 
PDF: f  x  
 exp   


s  2
 2  s  
1
 x  m 
1 
CDF: F  x    1  erf 
 
2 
s

2




where
Error Function: erf  x  
2

x
  
 exp t 2
0
The Normal distribution is denoted as X ~ N  m , s 2 
This probability density function (PDF) is

a symmetrical, bell-shaped curve,
centered at its expected value µ.

The variance is s2.

59
Normal distribution


Example
The simplest case of the normal distribution, known as the Standard Normal
Distribution, has expected value zero and variance one. This is written as
N(0,1).
60
Normal Distribution

Evaluating the distribution:


Independent of m and s, using the standard normal distribution:
Z ~ N  0,1
Transformation of variables: let Z 
X m
xm 

F ( x )  P  X  x   P Z 

s 

( xm ) /s
1 z2 / 2

e
dz

2

( xm ) /s

s
, where ( z )  
z

1 t 2 / 2
e
dt
2
 ( z )dz   ( xs m )
61
Normal Distribution

Example: The time required to load an oceangoing vessel, X, is
distributed as N(12,4)

The probability that the vessel is loaded in less than 10 hours:
 10  12 
F (10)  
   (1)  0.1587
 2 

Using the symmetry property, (1) is the complement of  (-1)
62
Other Distributions
Weibull Distribution
 Lognormal Distribution

63
Weibull Distribution

A random variable X has a Weibull distribution if its pdf has the form:
 b  x   b 1
  x   b 

exp  
 , x  
f ( x)  a  a 
a
 
 
0,
otherwise


3 parameters:
(    )

Location parameter: u,

Scale parameter: b , b  0

Shape parameter. a,  0

Example: u = 0 and a = 1:

Lifetime of objects
When b = 1,
X ~ exp(l = 1/a)
64
Lognormal Distribution

A random variable X has a lognormal distribution if its pdf has
the form:
 1
 ln x  μ  2 

exp 
, x  0
2
f ( x)   2π σx
2σ


0,
otherwise




m=1,
s2=0.5,1,2.
2
Mean E(X) = em+s /2
2
2
Variance V(X) = e2m+s /2 (es - 1)
Relationship with normal distribution



When Y ~ N(m, s2), then X = eY ~ lognormal(m, s2)
Parameters m and s2 are not the mean and variance of the lognormal
general reliability analysis
65
65
Empirical Distribution
66
Empirical Distributions

An Empirical Distribution is a distribution whose
parameters are the observed values in a sample of
data.
May be used when it is impossible or unnecessary to establish
that a random variable has any particular parametric
distribution.
 Advantage: no assumption beyond the observed values in the
sample.
 Disadvantage: sample might not cover the entire range of
possible values.

67
Empirical Distributions



In statistics, an empirical distribution function is a cumulative
probability distribution function that concentrates probability 1/n
at each of the n numbers in a sample.
Let X1, X2, …, Xn be iid random variables in with the CDF equal to F(x).
The empirical distribution function Fn(x) based on sample X1, X2, …, Xn
is a step function defined by
number of element in the sample  x
1
Fn  x  

n
n
n
 I X
i
x
i 1
1 if  X i  x 

I X i  x   

0 otherwise
where I(A) is the indicator of event A.
 For a fixed value x, I(Xi≤x) is a Bernoulli random variable with
parameter p=F(x), hence nFn(x) is a binomial random variable
with mean nF(x) and variance nF(x)(1-F(x)).
68
End of Chapter 4
69