Transcript Relation between Binomial and Poisson Distributions

Multinomial Distribution
• The Binomial distribution can be extended to describe number of
outcomes in a series of independent trials each having more than 2
possible outcomes.
• If a given trail can result in the k outcomes E1, E2, …, Ek with
probabilities p1, p2, …, pk, then the probability distribution of the
random variables X1, X2, …, Xk, representing the number of
occurrences for E1, E2, …, Ek in n independent trials is
n!
p X1 ,...,X k x1 ,..., xk  
p1x1 p2x2  pkxk
x1! x2! xk !
k
with
x
i 1
i
k
 n , and
p
i 1
i
 1.
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Poisson Processes
•
Recall, the Poisson random variable counts the number of events occurring in a
time (or space) interval where λ (a parameter of the distribution) is the rate of
the occurrence of the events per one unit of time (or space).
•
Very often we are interested in the number of events occurring in t units of
“time”, “space”, “area”, or “volume”.
•
The model for that case is the Poisson Process. It has the following properties:
1. The number of outcomes occurring in one time interval (or other specified
region) is independent of the number that occurs in any other disjoint time
interval. Process possessing this property is said to have no memory.
2. The probability that a single outcome will occur during a very short time
interval is proportional to the length of the time interval and does not
depend on the number of outcomes occurring outside this time interval.
3. The probability that more then one outcome will occur in such a short time
interval is negligible.
•
The probability distribution for the random variable that counts the number of
events per t units of time is given by…
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The Uniform distribution
•
X has a uniform[0,1] distribution. The pdf of X is given by:
1
f X x   
0
•
In general, if X has a Uniform[a, b] distribution, b > a. The pdf of X is
given by:
 1

f X x    b  a

0
•
0  x 1
otherwise
a xb
otherwise
The mean and variance of the Uniform distribution are ….
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The Exponential Distribution
• A random variable X that counts the waiting time for rare
phenomena has Exponential(λ) distribution. The parameter of the
distribution λ = average number of occurrences per unit of time
(space etc.). The pdf of X is given by:
e  x
f X x   
 0
x0
otherwise
• Questions: Is this a valid pdf? What is the cdf of X?
• Note: The textbook uses different parameterization λ = 1/β.
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Important Facts about Exponential Distribution
• The Exponential random variable possess and important property
called Memoryless property. It is described as follows:
P X  t  s | X  t   ...
• The Exponential distribution is often used to describe lifetime of
machines or other devices. (Read more on Section 6.7).
• The mean and Variance of the Exponential distribution are…
• The Exponential distribution very often describe the time until the
occurrence of a Poisson event or the time between Poisson events.
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More on Poisson Processes
• Model for times of occurrences (“arrivals”) of rare phenomena where
λ – average number of arrivals per one unit of time period.
X – number of arrivals in t units of time period.
x
e t t 
P X  x  
x!
, x  0,1, 2,...
• How long do I have to wait until the first arrival?
Let Y = waiting time for the first arrival (a continuous r.v.) then we have
PY  t   P X  0 on (0, t ]  e t
Therefore,
FY t   PY  t   1  e t
which is the CDF of the Exponential distribution.
• The waiting time for the first occurrence of an event when the number of
events follows a Poisson distribution is Exponentially distributed.
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The Gamma distribution
• A random variable X is said to have a gamma distribution with parameters
α > 0 and λ > 0 if and only if the density function of X is
 e  x x  1

f X  x     

0

where
0 x
otherwise

    e t t  1dt
0
• Note: the quantity г(α) is known as the gamma function. It is defined for
α > 0 and has the following properties:
– г(1) = 1
– г(α + 1) = α г(α)
– г(n) = (n – 1)! if n is an integer.
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