Binoial and Poisson Probability Distributions There are a few discrete robability distributions that cro u any ties in hysics alications, e.g. QM, SM. Here we consider TWO iortant and related cases, the Binoial & Poisson distributions. Binoial Probability Distribution Consider a situation where there are only two ossible outcoes a Bernoulli trial Exales: fliing a coin: The outcoe is either a head or tail rolling a die as a Bernoulli trial: 6 or not 6 i.e.,, 3, 4, 5 Label the ossible outcoes by the variable k We want to find the robability Pk for event k to occur Jaes Bernoulli Jacob I Since k can take on only values we define those values as: k = or k = Let the robability for outcoe k to occur be: Pk = = q reeber q soething ust haen so Pk = + Pk = = utually exclusive events Pk = = = - q We can write the robability distribution Pk as: Pk = k q -k Bernoulli distribution note: this forula satisfies all conditions of a robability distribution coin toss: define robability for a head as P Pk= =.5 and P=tail =.5 too! die rolling: define robability for a six to be rolled fro a six sided dice as Pk= Pk= = /6 and Pk==not a six = 5/6. born in Basel, Switzerland Dec. 7, 654-Aug. 6, 75 He is one 8 atheaticians in the Bernoulli faily. fro Wikiedia R.Kass/S5 P37 Lecture
How do we calculate the ean & variance? => use the definition of ean & variance! What is the ean of Pk? kpk Pk k k q q What is the Variance of Pk? k k P k P k k P Mean and Variance of a discrete distribution reeber +q= P Let s do soething ore colicated: Suose we have trials e.g. we fli a coin ties what is the robability to get successes = heads? Consider tossing a coin twice. The ossible outcoes are: no heads: P = = q one head: P = = q + q toss is a tail, toss is a head or toss is head, toss is a tail =q we don't care which h of fthe tosses is a head so two heads: P = = there are two outcoes that give one head ote: P=+P=+P==q + q + q + = +q = as it should! We want the robability distribution P,, where: = nuber of success e.g. nuber of heads in a coin toss = nuber of trials e.g. nuber of coin tosses = robability for a success e.g..5 for a head R.Kass/S5 P37 Lecture q
If we look at the three choices for the coin fli exale, each ter is of the for: C q - =,,, = for our exale, q = - always! coefficient C takes into account the nuber of ways an outcoe can occur without regard to order. for = or there is only one way for the outcoe both tosses give heads or tails: C = C = for = one head, two tosses there are two ways that this can occur: C =. Binoial coefficients: nuber of ways of taking things at tie! C,!! factorial:! =! =,! = =, 3! = 3 = 6,! = 3 Order of occurrence is not iortant e.g. tosses, one head case = we don't care if toss roduced the head or if toss roduced the head Unordered grous such as our exale are called cobinations Ordered arrangeents are called erutations For distinguishable objects, if we want to grou the at a tie, the nuber of erutations:! P,! exale: If we tossed a coin twice =, there are two ways for getting one head = exale: Suose we have 3 balls, one white, one red, and one blue. uber of ossible airs we could have, keeing track of order is 6 rw, wr, rb, br, wb, bw: 3! P 3, 3! 6 If order is not iortant rw = wr, then the binoial forula gives the nuber of cobinations: 3! C 3,!3! 3 nuber of two color cobinations R.Kass/S5 P37 Lecture 3
Putting it all together.. Binoial distribution: the robability of success out of trials:!! P,, C, q q q!! is robability of a success and q = - is robability of a failure.4.3 The binoial distribution changes shae deending on n, Exectation Value = n = 7 * /3 =.333....4.. Exectation Value = n = 5 * /3 = 6.667... P k, 7, /3. P k, 5, /3.8.6..4.. 4 6 8 k To show that t the binoial i distribution ib ti is roerly noralized, use Binoial i Theore: a b P,, k k k l l a k l b l q q binoial distribution is roerly noralized. 5 5 5 3 k R.Kass/S5 P37 Lecture 4
Mean of binoial distribution by definition: Mean & variance of Binoial Distribution q P P P,,,,,, A cute way of evaluating the above su is to take the derivative: q q q Variance of binoial distribution obtained using siilar trick: P,, R.Kass/S5 P37 Lecture 5,, P,, q
Exale: Suose you observed secial events success in a sale of events The easured robability efficiency for a secial event to occur is: What is the error standard deviation on the robability "error on the efficiency": q we will derive this later in the course The sale size should be as large as ossible to reduce the uncertainty in the robability easureent. Let s relate the above result to Lab where we throw darts to easure the value of. If we inscribe a circle inside a square with side=s then the ratio of the area of the circle to the rectangle is: r area of circle r s / area of square s s 4 So, if we throw darts at rando at our rectangle then the robability of a dart landing inside the circle is just the ratio of the two areas, /4. The we can deterine using:. The error in is related to the error in by: 4 We can estiate how well we can easure by this ethod by assuing that = 3.459 /4:.6 4 using / 4 This forula says that to irove our estiate of by a factor of we have to throw ties as any darts! Clearly, this is an inefficient way to deterine. R.Kass/S5 P37 Lecture 6 s
Exale: Suose a baseball layer's batting average is.3 3 for on average. Consider the case where the layer either gets a hit or akes an out forget about walks here!. rob. for a hit: =.3 rob. for "no hit : q = - =.7 On average how any hits does the layer get in at bats? = =.3 = 3 hits What's the standard deviation for the nuber of hits in at bats? =q / =.3.7 / 4.6 hits we exect 3 ± 5 hits er at bats Pete Rose s lifetie Consider a gae where the layer bats 4 ties: batting average:.33 robability of /4 =.7 4 = 4% robability of /4 = [4!/3!!].3.7 3 = 4% robability of /4 = [4!/!!].3.7 = 6% robability of 3/4 = [4!/!3!].3 3.7 = 8% robability of 4/4 = [4!/!4!].3 4.7 = % robability of getting at least one hit = - P = -.4=76% R.Kass/S5 P37 Lecture 7
Poisson Probability Distribution The Poisson distribution is a widely used discrete robability distribution. Consider a Binoial distribution with the following conditions: is very sall and aroaches exale: a sided dice instead of a 6 sided dice, = / instead of /6 exale: a sided dice, = / is very large and aroaches exale: throwing or dice instead of dice The roduct is finite Exale: radioactive decay radioactive decay nuber of Prussian soldiers kicked to death by horses er year! quality control, failure rate redictions Suose we have 5 g of an eleent very large nuber of atos: avogadro s nuber is large! Suose the lifetie of this eleent = years 5x 9 seconds robability of a given nucleus to decay in one second is very sall: = / = x - /sec BUT = /sec finite! The nuber of decays in a tie interval is a Poisson rocess. Poisson distribution can be derived by taking the aroriate liits of the binoial distribution P,,!! q using :!!! q! x a x xa a a!!! 3 x a a a... 3! >>! Siéon Denis Poisson June, 78-Aril 5, 84 e R.Kass/S5 P37 Lecture 8
P,, Let! e P, e! note : P, e! e! e e is always an integer does not have to be an integer The ean and variance of It is easy to show that: a Poisson distribution are the = = ean of a Poisson distribution sae nuber! = = = variance of a Poisson distribution Radioactivity exale with an average of decays/sec: i What s the robability of zero decays in one second?, e e e.35 3.5%! ii What s the robability of ore than one decay in one second?,,, e e e e.594 59.4%!! iii Estiate the ost robable nuber of decays/sec. P, * To attack this roble its convenient to axiize P, instead of P,. e P,!! There is no closed for exact solution for the that axiizes this function. We can get an aroxiate solutions using Stirling s aroxiation for n! R.Kass/S5 P37 Lecture 9
Stirling's Aroxiation:!!=5. -=3.3 4% 5!=48.48 55-5=45.6.9% P,! axiize the function by setting derivative to zero * The ost robable value for * is just the average of the distribution. Again, this is only aroxiate since Stirling s Aroxiation is only valid for large. A ore recise aroxiation for n! is n!=½[n+⅓ π]+n[n]-n However, this aroxiation yields the following equation for *: e * * /3 Strictly seaking can only take on integer values while is not restricted to be an integer. Finally, another iortant thing to reeber: If you observed events in a counting exerient, the error square root of variance on is This coes about because the nuber of events in a bin of a histogra is considered to be governed by a Poisson distribution and the ean and the variance of a Poisson distribution are the sae. R.Kass/S5 P37 Lecture
Coarison of Binoial i and Poisson distributions ib i with ean = ity Probabil 5.5.4 3.3.. oisson binoial =3, =/3 3 4 5 Probability 4.4.35.3 5.5 For large and fixed: Binoial Poisson..5..5 binoial =,=. oisson ot uch difference between the!... 3. 4. 5. 6. 7. R.Kass/S5 P37 Lecture
Unifor distribution & Rando ubers What is a unifor robability distribution function: x? x=constant c for a x b x=zero everywhere else Therefore x dx = x dx if dx =dx equal intervals give equal robabilities For a unifor distribution with a=, b=wehavex= x dx cdx c dx c What is a rando nuber generator? A nuber icked at rando fro a unifor distribution with liits [,] All ajor couter languages C, C++ coe with a rando nuber generator. In C++ rand gives a rando nuber. The following C++ rogra generates 5 rando nubers: #include<iostrea> #include<ath.h> #include<stdlib.h> int ain { // rogra to calculate and rint out soe rando nubers int K, oints; srand; //initial rando nuber generator oints=4; fork=;k<oints;k++ { std::cout<<" K "<<K<<" rando nuber "<<rand/.*rad_max<<"\n"; } return ; } K= rando nuber.8488 K= rando nuber.394383 K= rando nuber.78399 K= 3 rando nuber.79844 K= 4 rando nuber.9647 x If we generate a lot of rando nubers all equal intervals should contain the sae aount of nubers. For exale: generate: 6 rando nubers exect: 5 nubers [.,.] 5 nubers [.45,.55] With a rando nuber generator we can generate every other robability distribution! x R.Kass/S5 P37 Lecture
A C++ rogra to throw dice #include<iostrea> #include<cstdio> #include<ath.h> #include<stdlib.h> //The following line is used to run this rogra on the Physic Deartent s UIX couters e.g. FOX // coile and link unix using: cxx -o abc abc.c l where abc.c is the file that contains the source code int ain { // rogra to roll dice int nuroll;// nuber of rolls int K,dice; float roll[7]; //kees track of how often a,..6 occurs float rob; // robability of rolling a,,3..6 // srand; //initialize rando nuber generator std::cout<<"give nuber of rolls of the dice "<<"\n"; std::cin>>nuroll; std::cout<<"nuber of rolls of the dice "<< nuroll<<"\n"; fork=;k<=6;k++ { roll[k]=; } fork=;k<nuroll;k++ { // % is the odulus oerator dice =+rand%6; roll[dice]++; // cout<<"dice "<<dice<<"\n"; } fork=;k<=6;k++ { rob=roll[k]/nuroll; roll[k]/nuroll; std::cout<<"roll of dice "<<K<<" nuber "<<roll[k]<<" rob "<<rob<<"\n"; } return ; } R.Kass/S5 P37 Lecture 3