P r o b a b ilit y

P r o b a b ilit y D i s t r ib u t io n s q. Random Variables q. Experimental vs. Parent Distributions q. Binomial Distribution q. Poisson Distribution q. Gaussian Distribution

Random Variables https: //en. wikipedia. org/wiki/Random_variable “A random variable is a variable whose (measured) value is subject to variations due to chance…” A probability distribution describes the frequency of occurrence of a given value for a random variable Discrete Die Roll Continuous Time between PMT hits in a HAWC tank

● E xp e rim e n ta l v s Pa r e n t Ex p e r im e n D ta l: is. Ift r. I im n ns bauk te io 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

● ● E xp e rim e n ta l v s Pa r e n t Ex p e r im e n D ta l: is. Ift r. I im n ns bauk te io 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 xp e rim e n ta l v s Pa r e n t If I m a k e n m e a s u r e m e n ts o f a D is t r i b u t i o n s 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 ● hist o g r a m t o d e te rm ine 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 each b in b y the tota l n u m b er of 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 a

B i n o m ia l D i s t r ib u t i o n

http: //www 3. nd. edu/~rwilliam/stats 1/x 13. pdf X 13. 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 ds?

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 suffering from a d is e a s e f o r w h i c h rt h ae t e m oisr t a li 8 t y 0 %. F in d p r o b a b ili t i th e th a t (a ) e n oense( b ) t hse p a t ie n ts s u r v iv o wfo mor e xn ae cst u lyr v i v e s ( c ) t o or surviv e. 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 W rhwa at r. Id a n d 5 0 % b a c k w a r d. o b se rv e : K 472 back sca t t e r T 5 2 8 forw a rd sca t t e r 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 MEAN: If we perform an experiment N times, and ask how many successes are observed, the average number will approach the mean=m,

Derivation of Mean of Binomial Distribution http: //www. math. ubc. ca/~feldman/m 302/binomial. pdf

Derivation of Mean of Binomial Distribution Invoke Binomial Formula Use p+1 -p=1

Derivation of Variance of Binomial Distribution

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/Binomial. Distribution/

Binomial Distribution Matlab Demo http: //www. mathworks. com/help/stats/binomial-distribution. html

Po i s s o n D istribution http: //demonstrations. wolfram. com/Poisson. Distribution/

B in o m ia l D i s t r ib u ti on Po is s o n D i s t r ib u ti on

Derivation of Poisson Distribution

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 can take on d is c r e t e i n t e g e r v alues. ● n m ust b e a n i n te g e r ● need not be!

Ex a m p l e : Sup p o se th ere a r e 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 that one or 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 xp e rim e n t s (La b # 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. Neon Ca t h o d e (HV) Anod e (+ HV)

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. Neon - + + + Ca t h o d e (HV) Anod e (+ HV) 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 : Equip m e n t Sc h e m a t i c o s c illo s c o pe So u r c e G. M. ● H V 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 tvhorlteasgh o ld e u t p u t s d ig it a l p u ls e if V > O V Co m p a r a t or ● ● GM TH Sc a le r c o u n t s d ig it a l p u ls e s

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 hy do I exp ect the num ber of 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 on ● 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 re so m e p ra c t ical 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 ing so ? ● W o u ld a n in t e r p r e t a 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 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 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 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 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 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 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 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 istribution Mathematica Demo http: //demonstrations. wolfram. com/Poisson. Distribution/

Po i s s o n D i s t r i b u t i o n M a t l a b Demo http: //www. mathworks. com/help/stats/poissondistribution. html

B in o m ia l D is t r ib u t i on Po is s o n D i s t r ib u ti on 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/ss 09/Lecture 3. pdf Gauss. pptx

Ad dition al Reading an 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 ro b lem s: ● – 5. 4 , 5. 6 , 5. 1 2 1 0. 9 , 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 "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 e. g or or Binomial Coefficients The total number of combinations of k objects selected from a set of n different objects. http: //mathworld. wolfram. com/Binomial. Theorem. html