Transcript Review of Probability Models

1
A Review of
Probability Models
Dr. Jason Merrick
Bernoulli Distribution
The simplest form of
random variable.
0.7
– Success/Failure
– Heads/Tails
P ( X  1)  p
P ( X  0)  1  p
0.6
0.5
P(X=x)
•
0.4
0.3
0.2
0.1
0
E[ X ]  p
0
1
X
Var( X )  p(1  p )
Review of Probability Models
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Binomial Distribution
The number of successes
in n Bernoulli trials.
0.3
– Or the sum of n Bernoulli
random variables.
n x
P( X  x )    p (1  p )n  x
 x
0.2
P(X=x)
•
0.1
E[ X ]  np
0
Var( X )  np(1  p )
0
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Review of Probability Models
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Geometric Distribution
The number of Bernoulli
trials required to get the
first success.
0.7
0.6
0.5
P( X  x )  p x (1  p)n  x
1
E[ X ] 
p
Var( X ) 
(1  p )
p2
P(X=x)
•
0.4
0.3
0.2
0.1
0
0
1
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9
10
X
Review of Probability Models
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Poisson Distribution
The number of random
events occurring in a fixed
interval of time
– Random batch sizes
– Number of defects on an area
of material
P( X  x ) 
x
x!
E[ X ]  
e 
0.3
0.2
P(X=x)
•
0.1
0
0
Var( X )  
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Review of Probability Models
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Exponential Distribution
Model times between
events
–
–
–
–
Times between arrivals
Times between failures
Times to repair
Service Times
f ( x) 
1

e
x / 
E[ X ]  
0.5
0.4
0.3
f(x)
•
0.2
0.1
Var( X )   2
•
0
Memoryless
0
2
4
6
8
10
X
f ( x  y | X  y )  f ( x)
Review of Probability Models
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Erlang Distribution
•
The sum of k exponential
random variables
1
f ( x)  k
x k 1e  x / 
 (k  1)!
0.2
0.15
•
Var( X )  k 2
f(x)
E [ X ]  k
Gives more flexibility than
exponential
0.1
0.05
0
0
2
4
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X
Review of Probability Models
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Gamma Distribution
A generalization of the
Erlang distribution,  is
not required to be integer
f ( x) 
1
 1  x / 
x
e

 ( )
E [ X ]  
•
•
1.5
Var( X )   2
  0.5
1
f(x)
•
 1
0.5
 2
More flexible
Has exponential tail
0
0
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Review of Probability Models
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Weibull Distribution
•
•
Commonly used in
reliability analysis
The rate of failures is ( x /  )

1.5
  1 ( x /  )
f ( x)   x e

1
f(x)

E[ X ] 

  0.5
1
 
 
0.5
 2
2

   2  1   1  
Var( X )  2 2     
          
2
 1
0
Review of Probability Models
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Normal Distribution
The distribution of the
average of iid random
variables are eventually
normal
f ( x) 
E[ X ]  
•

1
2 2
e
1
2
2
0.45
 x   2
Var( X )   2
Central Limit Theorem
0.3
f(x)
•
0.15
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0
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Review of Probability Models
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Log-Normal Distribution
Ln(X) is normally
distributed.
– Used to model quantities that
are the product of a large
number of random quantities
– Highly skewed to the right.
1
(ln( x )   ) 2 / 2 2
f ( x) 
e
x 2
E[ X ]  e 
2
0.4
0.3
f(x)
•
0.2
0.1
/2
0
0
Var( X )  e
2   2
1
2
3
4
5
X
2
(e  1)
Review of Probability Models
C5/11
Triangular Distribution
Used in situations were
there is little or no data.
– Just requires the minimum,
maximum and most likely
value.
2( x  a )
f ( x) 
, axm
( m  a )( b  a )
2( b  x )

, m xb
(b  m )( b  a )
 0, otherwise
E [ X ]  ( a  b) / 2
0.3
0.2
f(x)
•
0.1
0
0
1
2
3
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X
Var( X )  (b  a )2 / 12
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Beta Distribution
Again used in no data
situations.
3
– Bounded on [0,1] interval.
– Can scale to any interval.
– Very flexible shape.
2.5
2
(   )  1
f ( x) 
x (1  x )  1
( )(  )
E[ X ] 
f(x)
•
1.5
1



Var( X ) 
(   )2 (    1)
0.5
0
0
0.25
0.5
0.75
1
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Review of Probability Models
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Homogeneous Poisson Process
• The number of events happening up to time t is
Poisson distributed with rate t
– The number of events happening in disjoint time intervals
are independent
– The time between events are then independent and
identically distributed exponential random variables with
mean 1/ 
– Combining two Poisson processes with rates  and  gives
a Poisson process with rate  + 
– Choosing events from a Poisson process with probability p
gives a Poisson process with rate p 
– A homogeneous Poisson process is stationary
Review of Probability Models
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Renewal Process
• If the time between events are independent and
identically distributed then the number of events
happening over time are a renewal process.
– The homogeneous Poisson process is a renewal process
with exponential inter-event times
– One could also choose the inter-event times to be Weibull
distributed or gamma distributed
– Most arrival processes are modeled using renewal
processes
– Easy to use as the inter-event times are a random sample
from the given distribution
– A renewal process is stationary
Review of Probability Models
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Non-stationary Arrival Processes
• External events (often arrivals) whose rate varies
over time
–
–
–
–
Lunchtime at fast-food restaurants
Rush-hour traffic in cities
Telephone call centers
Seasonal demands for a manufactured product
• It can be critical to model this nonstationarity for
model validity
– Ignoring peaks, valleys can mask important behavior
– Can miss rush hours, etc.
• Good model:
– Non-homogeneous Poisson process
Review of Probability Models
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Non-stationary Arrival Processes
(cont’d.)
• Two issues:
•
– How to specify/estimate the rate function
– How to generate from it properly during the simulation (will
be discussed in Chapters 8, 11 …)
Several ways to estimate rate function — we’ll just
do the piecewise-constant method
– Divide time frame of simulation into subintervals of time
over which you think rate is fairly flat
– Compute observed rate within each subinterval
– Be very careful about time units!
• Model time units = minutes
• Subintervals = half hour (= 30 minutes)
• 45 arrivals in the half hour; rate = 45/30 = 1.5 per minute
Review of Probability Models
C5/17