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Statistical Methods October 2008 Chapter 4 A
review of some standard discrete distributions
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Discrete distributions uniform binomial Poisson ge
ometric negative binomial
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• Uniform on 1, 2, , k
• models the occurrence of equally likely outcomes
• f(x) 1/k, x 1, 2, , k
• ? (k1)/2 , s2 (k2 1)/12
• e.g. score on a 6-sided die
• k 6, ? 3.5, s 1.71

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simulation of X uniform on 1,2,3,4,5,6 gt
xtrunc(6runif(1000) 1) gt table(x) x 1
2 3 4 5 6 162 164 160 154
179 181 gt summary(x) Min. 1st Qu. Median
Mean 3rd Qu. Max. 1.000 2.000
4.000 3.567 5.000 6.000 gt
sd(x) 1 1.729885
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Chart of frequencies
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• Binomial(n,p) X bi(n,p)
• models number of successes in n independent,
  identical trials with P(success) p at each
  trial
• G(t) pt (1 p)n
• ? np, s2 np(1 p)

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simulation of X bi(8, 1/6) gt xrbinom(1000,8,1/6
) gt table(x) x 0 1 2 3 4
5 213 386 267 106 23 5 gt summary(x)
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.000 1.000 1.000 1.355 2.000
5.000 gt sd(x) 1 1.032493
cf µ 1.33 , s 1.05
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Chart of frequencies

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pmf
symmetrical
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pmf
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pmf
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pmf
symmetrical
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pmf
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• Poisson(?) X P(?)
• models number of events which occur in a unit of
  time when events occur as a Poisson process
  with intensity/rate ?
• ? s2 ?

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simulation of X Poisson(1.5) gt
xrpois(1000,1.5) gt table(x) x 0 1
2 3 4 5 6 8 231 328 248 136
44 8 3 2 gt summary(x) Min. 1st
Qu. Median Mean 3rd Qu. Max. 0.000
1.000 1.000 1.482 2.000 8.000
gt var(x) gt sd(x) 1 1.489165 1
1.220314 cf µ 1.5 , s 1.22
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pmf
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pmf
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pmf
much more symmetrical
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• Geometric(p) X geo(p)
• number of trials/failures) before the first
  success occurs in a series of independent,
  identical trials with P(success) p at each
  trial
• ? q/p, s2 q/p2

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simulation of X geo(1/6) gt xrgeom(1000,1/6) pro
duced observations from 0 to 46, with mode 0,
median 4, mean 5.15, variance 29.95,
sd5.47
cf µ 5 , s 5.48
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histogram of simulated sample
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Negative binomial(k,p) X nb(k,p) number of
trials before the occurrence of the kth success
in a series of independent, identical trials with
P(success) p at each trial see Ch4 p4
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pmf
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FIN