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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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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
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= 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
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