Pr o b a b i l i t y
D i st r ib u tio n s
Random Variables
Experimental vs.
Parent Distributions
Binomial Distribution
Poisson Distribution
Gaussian Distribution
Random Variables
https://en.wikipedia.org/wiki/Random_variable
Discrete
“A random variable is a
variable whose
(measured) value is
subject to variations
due to chance…”
Die Roll
A probability distribution
describes the frequency
of occurrence of a given
value for a random
variable
Continuous
Time between PMT hits in a HAWC tank
E x p e r i m e n t a l v s Pa r e n t
D is t r i b u t io n s
●
Ex p e r im e n ta l: If I m a k e n
m e a s u r e m e n ts o f a q u a n t i t y x, t h e y
c a n b e so rt e d i n to a his t o g ra m t o
d e t e r m i n e t h e e x p e r im e n ta l
d is t r ib u t io n .
Physics 6719 Lecture 2
Physics 6719 Lecture 2
Physics 6719 Lecture 2
Physics 6719 Lecture 2
Physics 6719 Lecture 2
Physics 6719 Lecture 2
Physics 6719 Lecture 2
Physics 6719 Lecture 2
Physics 6719 Lecture 2
Physics 6719 Lecture 2
Physics 6719 Lecture 2
E x p e r i m e n t a l v s Pa r e n t
D is t r i b u t io n s
●
●
Ex p e r im e n ta l: If I m a k e n
m e a s u r e m e n ts o f a q u a n t i t y x, t h e y
c a n b e so rt e d i n to a his t o g ra m t o
d e t e r m i n e t h e e x p e r im e n ta l
d is t r ib u t io n .
If I d i v i d e t h e n u m b e r o f e v e n ts in
e a c h b in b y t h e t o t a l n u m b e r o f
e v e n ts , I h a v e a n e x p e r im e n t a l
p r o b a b il i t y d i s t r ib u t io n .
Physics 6719 Lecture 2
E x p e r i m e n t a l v s Pa r e n t
D is t r i b u t i o n s
If I m a k e n m e a s u r e m e n ts o f a q u a n t it y x,
t h e y c a n b e s o r t e d in t o a h i s t o g r a m t o
d e t e r m i n e t h e e x p e r im e n t a l d is t r ib u t io n .
●
If I d i v i d e t h e n u m b e r o f e v e n ts in e a c h b i n
b y t h e t o t a l n u m b e r o f e v e n ts , I h a v e a n
e x p e r im e n ta l p r o b a b i l i t y d is t r ib u t io n .
●
T h e p a r e n t p r o b a b il i t y d is t r ib u t io n is t h e
d is t r ib u t io n w e w o u l d s e e a s n → in f i n it y .
●
T h e p h y s ic s lies in t h e p r o p e r t ie s ( m e a n ,
w id t h ...) o f t h e p a r e n t d is t r ib u t io n , w h i c h
●
we must try to infer
B i n o m ia l D i s t r ib u t i o n
http://www3.nd.edu/~rwilliam/stats1/x13.pdf
X13.ppt
E x a m p l e : If I t o s s a c o i n 3
t i m e s , w h a t is t h e p r o b a b ili t y o f
o b t a in i n g 2 h e a d s ?
E x a m p l e : A h o s p ita l a d m its fo u r
p a t i e n ts
su ffering
from
a
d is e a s e f o r w h i c h t h e m o r t a li t y
ra te
is
80%.
F in d
th e
p r o b a b ili t i e s t h a t ( a ) n o n e o f
t h e p a t ie n ts s u r v iv e s ( b ) e x a c t ly
o n e su rv ive s (c) t w o or m o r e
surviv e.
Ex a m p le : In a s c a t t e r in g e x p e r im e n t, I c o u n t
fo r w a r d - a n d b a c k w a r d s c a tt e r i n g e v e n ts . I
e x p e c t 5 0 % fo r w a r d a n d 5 0 % b a c k w a r d .
W h a t I o b se rv e :
K
4 7 2 b ack scat t er
T
5 2 8 forw a rd sca t t er
W h a t u n c e r t a in t y s h o u l d I q u o t e ?
Mean of Binomial Distribution
Probability of getting n successes out of N tries,
when the probability for success in each try is p
N n
N n
PB (n , N ; p )    p 1  p 
n 
N
N!
  
n  v! N n !
MEAN: If we perform an experiment N times, and ask how many successes
are observed, the average number will approach the mean=m,
N n
m  lim n  PB (n , N ; p)  lim n   p 1  p N n  np
N 
N 
k 0
k 0
n 
N
N
Derivation of Mean of Binomial Distribution
N n
m  n  PB (n , N ; p)  n   p 1  p N n  Np
n 0
n 0
n 
N
N
N
N!
N n
m  E (n )  n 
pn 1  p 
n !N n !
n 0
N
N!
N n
m 
pn 1  p 
n  0 n  1! N n !
N
0! 1
N ! N ( N  1)! ( N  1)! ! (0  1)! 
N
0 0
N!
0  N! 0
N
N
n 0
p 0 1  p  
p 1  p   0
0  1!N !
0! N !
y  n  1; m  N  1

m  1! y 1
m y
m 
p 1  p 
y  0 y!m  y !
m
m
m  (m  1) p 
y 0
m!
m y
p y 1  p 
y!m  y !
http://www.math.ubc.ca/~feldman/m302/binomial.pdf
Derivation of Mean of Binomial Distribution
m
m  Np 
y 0
m!
m y
p y 1  p 
y!m  y !
Invoke Binomial Formula
a  b 
m
m

y 0
m!
a yb m y
y!m  y !
a  p; b  1  p
Use p+1-p=1
m

y 0
m
m!
m!
m y
m
m
y
p 1  p  
a y b m  y  a  b    p  1  p   1
y!m  y !
y  0 y!m  y !
 m  Np
Derivation of Variance of Binomial Distribution
  E (n 2 )  E (n ) 2  E (n 2 )  m 2
N
N n
E (n )  n  PB (n , N ; p )  n 2   pn 1  p 
n 0
n 0
n 
N
N
2
2
N
E (n (n  1))  n (v  1) 
n 0
N!
N n
pn 1  p 
n !N n !
N
N!
N n
pn 1  p 
n  2 n  2 ! N n !
E (n (n  1))  
E (n (n  1))  N ( N  1) p
N  2! pn 2 1  p N n

n  2 n  2 ! N n !
N
2
y  n  2; m  N  2
m
m!
m y
E (n (n  1))  N ( N  1) p 
p m 1  p 
y  0 m!m  y !
2
E (n (n  1))  N ( N  1) p 2 ( p  1  p )  N ( N  1) p 2
Derivation of Variance of Binomial Distribution
E (n (n  1))  N ( N  1) p 2 ( p  1  p )  N ( N  1) p 2
  E (n 2 )  E (n ) 2  E (n (n  1))  E (n )  E (n ) 2
E (n )  m  Np
  N ( N  1) p 2  Np  Np 2
  Np 2  Np 2  Np  Np 2  Np (1  p)
  Np (1  p)
Binomial Distribution Mathematica Demo
If a coin that comes up heads with probability p is tossed N times, the number of heads
observed follows a binomial probability distribution.
http://demonstrations.wolfram.com/BinomialDistribution/
Binomial Distribution Matlab Demo
http://www.mathworks.com/help/stats/binomial-distribution.html
Po i s s o n D i s t r i b u t i o n
http://demonstrations.wolfram.com/PoissonDistribution/
B in o m ia l
D i s t r ib u tio n
Po is s o n
D i s t r ib u tio n
Derivation of Poisson Distribution
PB (n , N ; p ) 
N!
N n
pn 1  p 
v! N n !
N!
N n
lim PB (n , N ; p )  lim
pn 1  p 
N 
N  v! N n !
N ( N  1)( N  2)  ( N  v  1) N n ! n
N n
p 1  p 
N 
v! N n !
lim PB (n , N ; p )  lim
N 
N ( N  1)( N  2)  ( N  v  1) n
N
n
p 1  p  1  p 
N 
v!
lim PB (n , N ; p )  lim
N 
m
lim    p
N  N
 
 N ( N  1)( N  2)  ( N  v  1)  m n   m   N   m   n 
lim PB (n , N ; p )  lim 
1     1     


N 
N 
v!
 N    N     N   

Derivation of Poisson Distribution
 N ( N  1)( N  2)  ( N  v  1)  m n   m   N   m   n 
lim PB (n , N ; p )  lim 
1     1     


N 
N 
v!
 N    N     N   

 mn
lim PB (n , N ; p )  
N 
 v!
N

  m 
  m 
N ( N  1)( N  2)  ( N  v  1)
 lim
 lim 1      lim 1    
n
N 
N 
N
  N   N   N  

n
N
  x 
  m 
e  lim1      lim 1      e  m
n 
N 
  n 
  N 
x
  m 
lim 1    
N 
  N 
n
n

m
 lim PB (n , N ; p )  e  m 
N 
 v!
 1n



n
Derivation of Poisson Distribution
N!
N n
pn 1  p 
v! N n !
N!
N n
lim PB (n , N ; p )  lim
pn 1  p 
N 
N  v! N n !
N ( N  1)( N  2)  ( N  v  1) N n ! n
N n
lim PB (n , N ; p )  lim
p 1  p 
N 
N 
v! N n !
N ( N  1)( N  2)  ( N  v  1) n
N
n
lim PB (n , N ; p )  lim
p 1  p  1  p 
N 
N 
v!
m
lim    p
N  N
 
 N ( N  1)( N  2)  ( N  v  1)  m n   m   N   m   n 
lim PB (n , N ; p )  lim 
1     1     


N 
N 
v!
 N    N     N   

PB (n , N ; p ) 
 mn
lim PB (n , N ; p )  
N 
 v!
N

  m 
  m 
N ( N  1)( N  2)  ( N  v  1)
 lim

lim

1



lim


n
 N 1   N  
N 
N 
N
N
  
  

n
N
  x 
  m 
e  lim1      lim 1      e  m
n 
N 
  n 
  N 
x
  m 
lim 1    
N 
  N 
n
 1n
 mn
 lim PB (n , N ; p )  e 
N 
 v!
m



n
Ex a m p le o f Po i s s o n D is t r i b u t i o n
●
●
●
P ois s o n d is t r ib u t e d
d a ta c a n t a k e o n
d is c r e t e i n t e g e r
v alues.
n m ust b e an
i n te g e r
mneed not be!
Ex a m p l e :
Sup p o se
th e r e
are
3 0 , 0 0 0 U n iv e r s it y o f U t a h s t u d e n ts ,
o f w h ic h 4 0 0 a r e p e r m it t e d t o c a r r y
g u n s . If I'm t e a c h in g a n a s t r o n o m y
c la s s o f 1 2 0 s t u d e n ts , w h a t is t h e
p r o b a b ilit y t h a t o n e o r m o r e is
c a r r y in g a g u n ?
E x a m p l e : C o u n t in g E x p e r i m e n t s
(Lab # 1)
G e ig e r -M ű lle r C o u n t e r
G e ig e r -M ű lle r C o u n t e r
N o b le g a s , e .g . N e o n
Ca t h o d e
(- H V)
Anode
(+ H V)
G e ig e r -M ű lle r C o u n t e r
N o b le g a s , e .g . N e o n
-
+
-
-
+
+
+
Ca t h o d e
(- H V)
Anode
(+ H V)
Io n iz in g p a r t ic le
G e ig e r -M ű lle r C o u n t e r : E q u i p m e n t
Sc h e m a t i c
o s c illo s c o p e
So u r c e
Co m p a r a t o r
G .M .
●
HV
●
●
Sc a le r
(“ co u n t e r ” )
“ Co m p a r a t o r ” c o m p a r e s G M
a n a lo g o u t p u t w it h t h r e s h o ld
v o lt a g e
O u t p u t s d ig it a l p u ls e if V > V
GM
Sc a le r c o u n t s d ig it a l p u ls e s
TH
E x p la in : U s in g a G e ig e r c o u n t e r , I m e a s u r e
t h e a c t i v it y o f a w e a k ly r a d io a c t iv e r o c k . I
r e c o r d a s m a ll n u m b e r ( < 5 ) c o u n ts in a
t e n s e c o n d in t e r v a l.
W h y d o I e x p e c t t h e n u m b e r o f c o u n t s I' d
m e a s u r e in r e p e a t e d t r i a ls t o b e Po is s o n
D is t r ib u t e d ?
D isc u s s i o n
●
●
●
Ca n a G e ig e r -d e t e c to r c o u n ti n g
e x p e r im e n t b e t r e a te d a s a binomial
distribution p r o b l e m ?
W h a t a r e s o m e p r a c t i c a l d i f f i c u l t ie s o n e
m ig h t e n c o u n t e r in d o i n g s o ?
W o u ld a n in t e r p r e ta t io n v ia t h e Poisson
distribution w o r k ?
W h a t H a p p e n s a s m B e c o m e s L a rg e ?
Physics 6719 Lecture 2
W h a t H a p p e n s a s m B e c o m e s L a rg e ?
Physics 6719 Lecture 2
W h a t H a p p e n s a s m B e c o m e s L a rg e ?
Physics 6719 Lecture 2
W h a t H a p p e n s a s m B e c o m e s L a rg e ?
Physics 6719 Lecture 2
W h a t H a p p e n s a s m B e c o m e s L a rg e ?
Physics 6719 Lecture 2
13 January
2012
49
W h a t H a p p e n s a s m B e c o m e s L a rg e ?
Physics 6719 Lecture 2
13 January
2012
50
W h a t H a p p e n s a s m B e c o m e s L a rg e ?
Physics 6719 Lecture 2
W h a t H a p p e n s a s m B e c o m e s L a rg e ?
Physics 6719 Lecture 2
13 January
2012
52
W h a t H a p p e n s a s m B e c o m e s L a rg e ?
Physics 6719 Lecture 2
W h a t H a p p e n s a s m B e c o m e s L a rg e ?
Physics 6719 Lecture 2
W h a t H a p p e n s a s m B e c o m e s L a rg e ?
Physics 6719 Lecture 2
13 January
2012
55
Po i s s o n D i s t r i b u t i o n
Mathematica Demo
http://demonstrations.wolfram.com/PoissonDistribution/
Po i s s o n D i s t r i b u t i o n M a t l a b D e m o
http://www.mathworks.com/help/stats/poissondistribution.html
B in o m ia l
D is t r ib u t i o n
Po is s o n
D i s t r ib u tio n
G a u s s ia n ( N o r m a l)
D is t r ib u t i o n
https://www.mpp.mpg.de/~caldwell/ss09/Lecture3.pdf
Gauss.pptx
A d d i t i o n a l R e a d i n g a n d Pr o b l e m s
●
R e a d in T a y l o r :
–
Ch 5 : T h e N o r m a l
Dist r ib u t io n
( Se c t io n s 1 a n d 2 )
–
Ch a p t e r 1 0 : T h e
B i n o m ia l
D i s t r ib u t i o n
–
Ch 1 1 : T h e
P o is s o n
Dist r ib u t io n
●
Tr y t h e p r o b l e m s :
–
5 .4 , 5 . 6 , 5 . 1 2
–
1 0 . 9 , 1 0 .1 0 , 1 0 .1 1 ,
1 0 . 2 0 , 1 0 .2 1 , 1 0 .2 2
–
1 1 .1 , 1 1 . 3 , 1 1 .8 ,
1 1 .1 0 , 1 1 . 1 4 , 1 1 .1 8 ,
1 1 .2 0
Binomial Expansion
x  y 4  x 4  4 x 3  6 x 2 y 2  4 xy 3  y 4
"Pascal's triangle 5" by User:Conrad.Irwin originally
User:Drini
Yang Hui triangle (Pascal's triangle) using
rod numerals, as depicted in a publication
of Zhu Shijie in 1303 AD.
https://en.wikipedia.org/wiki/Pascal's_rule
https://en.wikipedia.org/wiki/Yang_Hui
https://en.wikipedia.org/wiki/Pascal's_triangle
Blaise Pascal's version of the triangle
Binomial Formula for Positive Integral n
x  y n  x n  nx n1  n(n  1) x n2 y 2  n(n  1)(n  2) x n3 y 3    y n
x  y 
4
e.g
or
or
x  y 
n
x  y 
n
2!
3!
 x  4 x  6 x y  4 xy 3  y 4
4
3
2
2
 n  n 1  n  n  2 2  n  n 3 3
n n
 x    x    x y    x y      y
1 
2
3 
n
n
 n  k nk
    x y
k 0  k 
n
n
n
n
n
  y 
y
n
 x1     x 1       x k y n  k
 x
k 0  k 
  x 
Binomial Coefficients
n
n
n
 n  k n k n
 n  x 
1  n  n  k nk  n  n  k n nk
 y








  
1


x
y

x
y

x
/
y






n  
k 
k 

x  k 0  k 
 x
k

0
k

0
k

0
 
 
 k  y 



The total number of combinations of k objects selected from a set of n different objects.
 n  n(n  1)(n  2)  (n  k  1)
n 
n!
  


 
k!
k!(n  k )!  n  k 
k 
http://mathworld.wolfram.com/BinomialTheorem.html
k n