binomial and poisson distribution binomial and poisson distribution topic 7
TRANSCRIPT
![Page 1: Binomial and Poisson Distribution Binomial and Poisson Distribution Topic 7](https://reader036.vdocuments.mx/reader036/viewer/2022081419/56649efc5503460f94c10665/html5/thumbnails/1.jpg)
![Page 2: Binomial and Poisson Distribution Binomial and Poisson Distribution Topic 7](https://reader036.vdocuments.mx/reader036/viewer/2022081419/56649efc5503460f94c10665/html5/thumbnails/2.jpg)
)()(),( ly,Equivalent yYPxXPyYxXP
![Page 3: Binomial and Poisson Distribution Binomial and Poisson Distribution Topic 7](https://reader036.vdocuments.mx/reader036/viewer/2022081419/56649efc5503460f94c10665/html5/thumbnails/3.jpg)
, of averagerun -long theas dinterprete becan
and ofmean population thecalled also is )(
X
XXE
mean sample then the,,...,,
obtain to)(on distributi the toaccording
tlyindependen and repeatedly sample weif i.e.,
21 nXXX
xp
increases. as )()...( 11 nXEXXX nnn
![Page 4: Binomial and Poisson Distribution Binomial and Poisson Distribution Topic 7](https://reader036.vdocuments.mx/reader036/viewer/2022081419/56649efc5503460f94c10665/html5/thumbnails/4.jpg)
t.independen are and if )()()( 4.
e,Furthermor
YXYEXEXYE
![Page 5: Binomial and Poisson Distribution Binomial and Poisson Distribution Topic 7](https://reader036.vdocuments.mx/reader036/viewer/2022081419/56649efc5503460f94c10665/html5/thumbnails/5.jpg)
)var()( :deviation Standard XXSD
increases. size sample theas
)var( variancesample the
mean, theof case in the As2 XS
![Page 6: Binomial and Poisson Distribution Binomial and Poisson Distribution Topic 7](https://reader036.vdocuments.mx/reader036/viewer/2022081419/56649efc5503460f94c10665/html5/thumbnails/6.jpg)
Binomial and Poisson Binomial and Poisson
DistributionDistribution
Binomial and Poisson Binomial and Poisson
DistributionDistribution
Topic 7Topic 7
![Page 7: Binomial and Poisson Distribution Binomial and Poisson Distribution Topic 7](https://reader036.vdocuments.mx/reader036/viewer/2022081419/56649efc5503460f94c10665/html5/thumbnails/7.jpg)
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.
![Page 8: Binomial and Poisson Distribution Binomial and Poisson Distribution Topic 7](https://reader036.vdocuments.mx/reader036/viewer/2022081419/56649efc5503460f94c10665/html5/thumbnails/8.jpg)
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)
![Page 9: Binomial and Poisson Distribution Binomial and Poisson Distribution Topic 7](https://reader036.vdocuments.mx/reader036/viewer/2022081419/56649efc5503460f94c10665/html5/thumbnails/9.jpg)
M o r e g e n e r a l l y , t h e r e a r e
)!(!!
knkn
k
n
w a y s o f o r d e r i n g k B ’ s a n d n - k G ’ s . S o
P r ( k b o y s o u t o f n c h i l d r e n )
knk
GGBBk
n
)......Pr(
knk ppk
n
)1(
![Page 10: Binomial and Poisson Distribution Binomial and Poisson Distribution Topic 7](https://reader036.vdocuments.mx/reader036/viewer/2022081419/56649efc5503460f94c10665/html5/thumbnails/10.jpg)
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
![Page 11: Binomial and Poisson Distribution Binomial and Poisson Distribution Topic 7](https://reader036.vdocuments.mx/reader036/viewer/2022081419/56649efc5503460f94c10665/html5/thumbnails/11.jpg)
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)
1683.0
)125.01(125.02
7]2[ 52
XP
![Page 12: Binomial and Poisson Distribution Binomial and Poisson Distribution Topic 7](https://reader036.vdocuments.mx/reader036/viewer/2022081419/56649efc5503460f94c10665/html5/thumbnails/12.jpg)
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
![Page 13: Binomial and Poisson Distribution Binomial and Poisson Distribution Topic 7](https://reader036.vdocuments.mx/reader036/viewer/2022081419/56649efc5503460f94c10665/html5/thumbnails/13.jpg)
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 .
!
)1(k
epp
k
n kknk
)(~ PoissonY i f
!)(
k
ekYP
k
f o r ,...2,1,0k
i s c a l l e d t h e r a t e . O f t e n u s e d t o m o d e l c o u n t d a t a s u c h a s a ) # i n d u s t r i a l a c c i d e n t s i n a p l a n t p e r m o n t h b ) # c h r o m o s o m e i n t e r c h a n g e s w i t h i n c e l l s c ) # i n s u r a n c e c l a i m s
![Page 14: Binomial and Poisson Distribution Binomial and Poisson Distribution Topic 7](https://reader036.vdocuments.mx/reader036/viewer/2022081419/56649efc5503460f94c10665/html5/thumbnails/14.jpg)
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)
265.0!2
3.2)2(
23.2 eYP
X = # deaths in 1 year ~ Poisson(4.6)
1063.0!2
6.4)2(
26.4 eXP