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Discrete Distributions
UNIVERSITY OF PRETORIA
DEPARTMENT OF INDUSTRIAL AND SYSTEMS ENGINEERING
ENGINEERING STATISTICS BES221
DISCRETE DISTRIBUTIONS
The Chancellor of the Exchequer is a man whose
duties make him more or less of a taxing machine.
He is entrusted with a certain amount of misery which
it is his duty to distribute as fairly as he can.
Robert Lowe, Viscount Sherbrooke,
British Liberal politician
CONCEPTS
Random variables – Variates
Output from a stochastic process
Discrete and continuous variates and distributions
Histograms and distributions
Stochastic models – mathematical
BINOMIAL DISTRIBUTION
Variate : Number of “successes” in n Bernoulli trails
Number of “successes” in a sample of size n taken from
a large population
Jean (or Johann) Bernoulli (1667 – 1748)
Jacques (or Jakop) Bernoulli (1654 – 1705)(brother of Jean)
Daniel Bernoulli (1700 – 1782)(son of Jean)
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Discrete Distributions
Assumptions :
1. Only two possible outcomes – “success” and “failure”
2. Probability of a “success” stays the same and constant = p
3. There are n trails with n a constant
4. Trails are mutually independent
P(X=x) = b(x;n,p) = nCx px (1-p)n-x
x = 0,1,2,…, n
0p1
n > 0 and integer
Cumulative probabilities
Binomial probability tables
Symmetrical and skewed Binomial distributions
HYPERGEOMETRIC DISTRIBUTION
Sampling with and without replacement
Finite and infinite populations
All Binomial assumptions accept trails are not independent and
probabilities do not stay the same
Sample of size n taken from a population consisting of N items of
which a are “successes”
P(X=x) = h(x;n,a,N) = {aCx}{N-aCn-x} / {NCn}
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Discrete Distributions
POISSON DISTRIBUTION
Siméon-Denis Poisson (1781 – 1840) (Fr : Fish)
The Poisson process
Assumptions
1. The probability of a “success” during a small interval of
time is proportional to the length of the time interval
2. The probability of more than one “success” during a small
interval of time is negligible
3. The probability of a “success” during a small interval of
time does not depend on what happened during any of the
previous time intervals –no memory property
Variate : A discrete number of occurrences per time/space/entity
(Goals per game, Drop-goals per game, Typing errors per page,
Defectives per assembly, Arrivals per hour, Inclusions per m3,
Particles per liter, Breakdowns per day, Drop-outs per class,
Accidents per kilometer, Sixes per innings, Freckles per nose,
etc.)
P(X=x) = f(x;) = x e- / x!
for
x = 0,1,2, …
Poisson probability tables
Use Poisson as a good approximation to the Binomial if n  20
and p  0,05 or if n  100 then np  10
GEOMETRIC DISTRIBUTION
All the same assumptions as the binomial but the variate is the
number of the trail on which the first “success” occurs
P(X=x) = g(x;p) = p (1-p)x-1
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for
3
x = 1,2,3, …
Engineering StatisticsBES221
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Discrete Distributions
MULTI-NOMIAL DISTRIBUTION
Multi-variable distributions
Joint probability distributions
Marginal distributions
All the same assumptions as the Binomial except that each trail
may have k > 2 possible outcomes
f(x1, x2, …. xk) = {(n!) / ( x1! x2! …. xk!)} p1x1 p2x2 …. pkxk
(Number of oranges classified as small, medium or large in
sample)
CHEBYSHEV’S THEOREM
Pafnutti Lvovich Chebyshev (1821 – 1894)
 - k

 + k
X
P(X - )  1 / k2
Law of large numbers
Regression to the mean
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Discrete Distributions
MEAN AND VARIANCE OF DISTRIBUTIONS
Every distribution has a mean and variance which are functions of
the parameters of the distribution
In general :
 = Σ x f(x)
2 = Σ (x - )2 f(x)
Binomial :
=np
2 = n p (1-p)
Hypergeometric :  = n (a / N)
2 = {n a (N – a) (N – n)} / {N2 (N – 1)}
Poisson :
=
2 = 
There are many other somewhat specialized discrete distributions
The most widely used, and therefore most important, is the
Binomial and Poisson distributions
MONTE CARLO SIMULATION
By generating and transforming random numbers, using a pseudorandom generator, many statistical experiments may be simulated
with the help of a computer and appropriate software
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