Día 13 de mes: Comodín

STATISTICAL METHODS FOR BUSINESS
UNIT 3. DISCRETE PROBABILITY MODELS
3.1.- Bernoulli processes and related distributions.
3.1.1- Binomial model.
3.1.2- Geometric model.
3.2.- Hypergeometric model.
3.3.- Poisson model.
Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Statistical Methods for Business
UNIT 3. OBJECTIVES
• To identify key discrete probability models,
realizing the assumptions they are based
on.
• To handle expressions of the expectation
and the variance of the main models.
• To calculate probabilities for the main
models.
Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Statistical Methods for Business
PROBABILITY MODELS
• The behaviour of different random variables may exhibit
common patterns in practice.
• A probability model is a family of probability
distributions indexed by a common set of parameters.
UNIFORM MODEL
a
NORMAL MODEL
b
Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Statistical Methods for Business
Bernoulli distribution
Random experiment
Success
Random variable
Failure
1
Probability
0
P(X=1)=p P(X=0)=q
A SINGLE TRIAL
p
q = 1- p
Characteristics:
E(X)=p
0
1
X
Var(X)=pq
Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Statistical Methods for Business
Bernoulli Process
Each trial is a random experiment with two
possible outcomes
 n independent trials are carried out.
The n observations are mutually independient.
The probability of success (p) is the same for all the
trials.
The assumptions of Bernoulli processes allow us to
define
several
probability
models:
Binomial,
Geometric, …
Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Statistical Methods for Business
Bernoulli Processes
EXPERIMENT
SUCCESS
PROBABILITY
OF SUCCESS
NUMBER OF
TRIALS
Toss of a die
“To get a 2”
1
p=
6
10
Study on job
status
“Employed” Employement
rate
Poll before a
referendum
Call for an
interview
“To vote
YES”
“To attend”
30
p=0.4
240
p
20
“Number of successes in
n independent trials”
B(n, p)
Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Statistical Methods for Business
Binomial model
SUCCESS
RV
"k successes in n trials" XB(n, p)
Probability
P(X=k)
No. Of sequences of n
Trials with k successes
C n,k
k successes & (n-k)
k
n− k
p (1− p )
=“success"
="failure"
()
n k
n− k
P ( X=k )= p (1− p)
k
Binomial distribution
X: Number of successes in n trials.
Probability function
XB(n,p)
n k
n k
P  X  k     p 1  p 
k
k  0,1,...,n
Characteristics:
E ( X )=np
Var( X )=npq
Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Statistical Methods for Business
Some probability functions
XB(n=6, p=0.2)
XB(n=6, p=0.5)
XB(n=15, p=0.2)
XB(n=15, p=0.5)
Geometric model
EVENT
“No. of trials until the 1st. Success"
(k-1) times
RV
X  G(p)
Probability
P(X=k)
and 1
k−1
(1− p) p
=“success"
="failure"
Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Statistical Methods for Business
Geometric distribution
X: Number of trials until the first success.
P ( X=k )=(1− p)
k−1
p
XG(p)
k=1, .. . ,n,. ..
1
E ( X )=
p
q
Var ( X )= 2
p
Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Statistical Methods for Business
Hypergeometric model
=“sucess"
="failure"
X “Number of
in the sample”
How many samples of size n?
C N,n
M
N-M
n
How many samples of size n
contain k success?
C M,k C N− M,n−k
P ( X=k )=
C M,k C N− M,n−k
SAMPLE (without replacement!)
C N,n
=
( )( )
()
M
k
N −M
n−k
N
n
Hypergeometric distribution



 M  N  M



k n k 
Probability function
P(X  k )      
 N


n 
k= máx0, n-(N-M), …, mín(M,n)
Characteristics:
M
E ( X )=n =np
N
M ( N −M )( N −n)
( N −n)
Var ( X )=n
=npq
2
( N −1)
N ( N −1)
Convergence
H ( N,M,n )→ B(n,p )
Correction
factor
N > 50 and n <0.1N
Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Statistical Methods for Business
Poisson model
 No. of people living more than 100 years.
 No. of plane accidents.
 No. of errata on a page.
 No. of telephones dialed incorrectly on a day.
Characteristics
•
STABLE PROCESS ( = average number of events per
time unit)
•
INDEPENDENT EVENTS
•
PROBABILITY OF 2 OR MORE EVENTS IN A SHORT
INTERVAL  0
Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Statistical Methods for Business
Poisson distribution
X “Number of events in a certain interval”
Probability function
X  P()
 k
e 
P(X  k) 
k!
k= 0, 1, 2,…, n, ...
Characteristics
E(X) = 
Var(X) = 
Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Statistical Methods for Business