Probability Distributions  Random Variables  Experimental vs. Parent Distributions  Binomial Distribution  Poisson Distribution  Gaussian Distribution

Random Variables Die Roll Discrete Continuous Time between PMT hits in a HAWC tank “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

Experimental vs Parent Distributions ● Experimental: If I make n measurements of a quantity x, they can be sorted into a histogram to determine the experimental distribution.

Physics 6719 Lecture 2

Experimental vs Parent Distributions ● Experimental: If I make n measurements of a quantity x, they can be sorted into a histogram to determine the experimental distribution. If I divide the number of events in each bin by the total number of events, I have an experimental probability distribution. ●

Physics 6719 Lecture 2

Experimental vs Parent Distributions ● If I make n measurements of a quantity x, they can be sorted into a histogram to determine the experimental distribution. If I divide the number of events in each bin by the total number of events, I have an experimental probability distribution. The parent probability distribution is the distribution we would see as n → infinity. The physics lies in the properties (mean, width...) of the parent distribution, which we must try to infer ● ● ●

Binomial Distribution

X13.ppt

Example: If I toss a coin 3 times, what is the probability of obtaining 2 heads?

Example: A hospital admits four patients suffering from a disease for which the mortality rateis probabilities 80%.Findthe that(a)noneof the patients survives (b) exactly onesurvives(c)twoor survive. more

Example: In a scattering experiment, I count forward- and backward scattering events. I expect 50% forward and 50% backward. What I observe: T K 472 back scatter528 forward scatter What uncertainty should I quote?

Mean of Binomial Distribution Probability of getting 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= ,

Derivation of Mean of Binomial Distribution

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

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.

Binomial Distribution Matlab Demo

Poisson Distribution

Binomial Distribution Poisson Distribution

Derivation of Poisson Distribution

Example of Poisson Distribution ● Poisson distributed data can take on discrete integer values. n must be an integer  need not be! ● ●

Example: Suppose there are 30,000 University of Utah students, of which 400 are permitted to carry guns. If I'm teaching an astronomy class of 120 students, what is the probability that one or more is carrying a gun?

Example: Counting Experiments (Lab #1)

Geiger-Műller Counter

Noble gas, e.g. Neon Cathode (- HV) Anode (+ HV)

Geiger-Műller Counter Noble gas, e.g. Neon Cathode (- HV) Anode(+ HV)Anode(+ HV) Ionizing particleIonizing particle

Geiger-Műller Counter: Equipment Schematic HVHV SourceSource G.M.G.M. ComparatorComparator oscilloscopeoscilloscope Scaler (“counter”) ● “Comparator” compares GM analog output with threshold voltage ● Outputs digital pulse if V> V GMTH Scaler counts digital pulses ●

Explain: Using a Geiger counter, I measure the activity of a weakly radioactive rock. I record a small number (<5) counts in a ten second interval. Why do I expect the number of counts I'd measure in repeated trials to be Poisson Distributed?

Discussion ● Can a Geiger-detector counting experiment be treated as a binomial distribution problem? What are some practical difficulties one might encounter in doing so? Would an interpretation via the Poisson distribution work? ● ●

Physics 6719 Lecture 2 What Happens as  Becomes Large?

Physics 6719 Lecture 2 What Happens as  Becomes Large?

Physics 6719 Lecture 2 What Happens as  Becomes Large?

Physics 6719 Lecture 2 What Happens as  Becomes Large?

Physics 6719 Lecture 2 13 January What Happens as  Becomes Large?

Physics 6719 Lecture 2 13 January What Happens as  Becomes Large?

Physics 6719 Lecture 2 What Happens as  Becomes Large?

Physics 6719 Lecture 2 13 January What Happens as  Becomes Large?

Physics 6719 Lecture 2 What Happens as  Becomes Large?

Physics 6719 Lecture 2 What Happens as  Becomes Large?

Physics 6719 Lecture 2 13 January What Happens as  Becomes Large?

Poisson Distribution Mathematica Demo

Poisson Distribution Matlab Demo distribution.html

BinomialDistributionBinomialDistribution Poisson Distribution Gaussian (Normal) Distribution

Gauss.pptx

Additional Reading and Problems ● Read in Taylor: – Ch 5: The Normal Distribution (Sections 1 and 2) – Chapter 10: The Binomial Distribution – Ch 11: The Poisson Distribution ● Try the problems: – 5.4, 5.6, 5.12 – 10.9, 10.10, 10.11, 10.20, 10.21, – 11.1, 11.3, 11.8, 11.10, 11.14, 11.18, 11.20

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. rod numeralsZhu Shijie Blaise Pascal's version of the triangle

Binomial Formula for Positive Integral n or Binomial Coefficients or The total number of combinations of k objects selected from a set of n different objects. e.g