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Binomial and Poisson Binomial and Poisson

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Topic 7Topic 7

Binomial Probability Distribution

Consider a sequence of independent events with only two possible outcomes called success (S) and failure (F)Example: outcome of treatment (cured/not cured)

opinion (yes/no) S=yes, F=nogender (boy/girl) S=boy, F=girl

Let p be the probability of S Consider n number of such independent events.Then the total no of success out of n such events is a random variable called Binomial r.v.

Binomial Probability Distribution

Let p = probability of having a boy q = probability of having a girl

Assume the outcome of the first child (boy or girl) does not affect the outcome of the second and subsequent children i.e. events are independent ,we can multiply the probabilities together to get

Pr(BBB) = p x p x p = Pr(3 boys)

Pr(GGG) = q x q x q = Pr(No boy)

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Binomial Probability

For a 3-child family,1 boy = BGG or GBG or GGBPr(1 boy) = Pr(BGG)+Pr(GBG)+Pr(GGB) = pqq + qpq +qqp = 3pqq

Similarly,Pr(2 boys) = Pr(BBG)+Pr(BGB)+Pr(GBB) = 3ppq

Example 1Choose a group of 7 old individuals randomly from the population of 65-74 years old in US. Suppose 12.5% of the population in that age is diabetic. The total no. of persons out of the 7 selected who suffers from Diabetes has a binomial distribution. Let X = # diabetic Binomial( n = 7, p = 0.125)

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Example 2Past records indicate that 70% of patients responded to treatment. What is the probability that 16 out of the next 20 patients will respond to treatment?

# patients responding ~ Bin( n = 20, p = 0.7)

P(16 out of 20 responded)

416 )7.01(7.016

20

= 0.1304

P o i s s o n d i s t r i b u t i o n : C o n s i d e r a b i n o m i a l r a n d o m v a r i a b l e Y w i t h n v e r y l a r g e a n d p s m a l l a n d n p i s m o d e r a t e e q u a l t o . T h e n t h e p r o b a b i l i t i e s c a n b e a p p r o x i m a t e d b y w h a t i s c a l l e d a P o i s s o n r a n d o m v a r i a b l e .

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The number of deaths attributable to typhoid fever follows a Poisson distribution at a rate of 4.6 deaths per year

Y = # deaths in 6 months ~ Poisson(2.3)

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X = # deaths in 1 year ~ Poisson(4.6)

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