12 - Probability and Statistics in Physics
▮
Week of March XX

Outline
12.1 Probability Distributions
12.2 Three most important probability distributions
12.3 Example : Radioactive decay of radium
12.4

Import[]

Probability and Statistics in Physics

Statistics is the branch of mathematics used in the analysis of experimental data.

For example, “statitics” is necesary to account the effects of experimental errors

Also, “statistics” provides the methods for comparing data and theory.

/1/ Probability Distributions

“Statistics” involves the analysis of “random processes”.

Examples of random processes are flipping a coin, or throwing dice, or radioactive decays.

The outcomes of a random process, repeated many times, gives a set of experimental data.

Let X be a measurement, i.e., the quantitative observation in a random process. 
X is called a “random variable”.  X may be continuous or discrete.

Associated with the random process, there a probability distribution P(x).
▮ If X is a discrete variable, then P(x) is the probability that X will have the value x; the sum of P(x) must be 1.
▮ If X is a continuous variable, then P(x) dx is the probabiity that the measured value of X will lie in the interval (x,x+dx); the integral of P(x) must be 1.

Now we need to review some functional analysis of the probability distribution P(x).

Cumulative distributions

We might want to know the probability of finding X between certain limits,
e.g., x1 < X < x2. This probability is given by the cumulative,
or integrated distribution,

(1)

In Eq. (1) we are assuming that P(x) is a function of a continuous variable.
If the observable X takes discrete values then the integral is replaced by a sum,

(2)

The probability distribution is normalized to 1,

(3)

In words, Eq. (3) states that the probability is 1 that an observation will yield one of the possible outcomes; in other words, the observation must yield one or the possible outcomes.

Expectation Values

The expectation value of the random variable x, or a random variable function f(x), is defined by

(4)

So E[x] is the expected value of x = the average of many measurements.

As usual the discrete case is analogous, but replacing the integrals by summations.

The Mean and Variance and Standard Deviation

These quantities are familiar in data analysis.

◼ Sample Mean and Theoretical Mean

◼ Sample Variance and Theoretical Variance

◼ Standard Deviation

/2/ Three probability distributions most important in Physics

The three most important probability distributions in Physics are ...

	the binomial distribution (or, Bernoulli distribution) ;

	the Poisson distribution ;

	the Gaussian distribution (or, normal distribution) .

2.1 The Binomial Distribution

Consider a random process in which each result has only two possible values,
x = 1 (“success”) or x = 0 (“failure”).

Assume : probability of success = p	and	probability of failure = q = 1 – p.

Then what is the probability of r successes in N trials?

This probability is given by the binomial distribution.

If we assume that the probability of success (p) does not change from one trial to the next,   then the probability of getting success (i.e., x = 1)
r times in N trials is

where p is the probability of success in a single trial.

EXERCISE  1
In Mathematica the probability distribution for the binomial distribution P(r) is denoted by

	PDF [ BinomialDistribution [n , p] , r ] 

where p = the probability of success in one trial,
n = the number of trials,
r = the number of successes.

(A) flipping an unbiased coin
Let n = 20 trials and p = 0.5. Plot P(r) versus r for all possible values of r.
(Since r has only discrete values, r ∈ {1 2 3 ... n}, plot the function with the command DiscretePlot (see the help file if necessary).
Also, print the mean and variance, with the command Print.

(B) flipping a biased coin
Let n = 20 trials and p = 0.7 (i.e., the result x = ‘heads’ is more likely than x = ‘tails’.) Plot P(r) versus r for all possible values of r.
Also, print the mean and variance, with the command Print.

These commands will be useful ...

?PDF
?BinomialDistribution
?Mean
?Variance

Pa[r_]=PDF[BinomialDistribution[20,0.5],r];
Pb[r_]=PDF[BinomialDistribution[20,0.7],r];
pla=DiscretePlot[Pa[r],{r,0,20},
PlotStyle->{Red,Thickness[0.01]},ExtentSize->0.5,
Frame->True,FrameLabel->{"r","P(r)"},ImageSize->Medium]
Print["mean = ",Mean[BinomialDistribution[20,0.5]]]
Print["variance = ",Variance[BinomialDistribution[20,0.5]]]
plb=DiscretePlot[Pb[r]//Evaluate,{r,0,20},
PlotStyle->{Red,Thickness[0.01]},ExtentSize->0.5,
Frame->True,FrameLabel->{"r","P(r)"}]
Print["mean = ",Mean[BinomialDistribution[20,0.7]]]
Print["variance = ",Variance[BinomialDistribution[20,0.7]]]

2.2 The Poisson Distribution

There are several ways to derive the Poisson distribution.

First, the Poisson distribution is a limiting form of the binomial distribution.
Suppose in a Bernoulli process the probability p ⟶ 0 and the number of trials N ⟶ ∞, such that the mean µ = Np, remains constant.

The probability of observing r “successes” in this limit approaches

This is the Poisson distribution; r is the number of “successes”.

The figure illustrates the limit, by comparing three cases where the theoretical mean is 7. The red plots are binomial distributions with μ=7 and p = 0.7, 0.35, and 0.14; the black plots are the Poisson distribution with mean μ.

SetDirectory["/home/stump/Dated/03-March/LESSON12"]
Import["BinPoi.png","PNG"]

Note that the Poisson distribution is a discrete distribution.
The domain of P(r) is the set of non-negative integers, Z.

The Poisson distribution can describe processes for which the single-trial probability of success is very small (p⟶ 0) but in which the number of trials is so large (N⟶∞) that there is a nonzero number of success events.

Another way to derive the Poisson distribution is to consider a sequence of events with event rate = Γ, observed in time intervals  Δt.

A random process for which the probability distribution is the Poisson distribution is called a Poisson process.

Another derivation of the Posson distribution, based on radioactive decays

Information copied from an unidentified web site (no copyright given)

Poisson processes

A Poisson process is one in which occurrences are randomly distributed in time, space or some other variable with the number of occurrences in any non-overlapping intervals statistically independent.

For example, naturally occurring gamma rays detected in a scintillation detector are randomly distributed in time, or chocolate chips in a cookie dough are randomly distributed in volume. For simplicity, we will limit our discussion to occurrences or “events" randomly distributed in time. The “events” will be decays of unstable particles in a radioactive sample.

The average decay rate will be denoted Γ;
then in any time interval Δt the expected number of decays is Γ Δt ≡ μ.

While the expected number of decays for a given experiment need not be an integer, the number of number n actually observed must be. Moreover, due to the randomness of the decays, n may be more or less than Γ Δt.

We will prove that the probability to observe n events in the time interval Δt is

	P( n ) = (μ^n) exp(-μ ) / n! 

where μ = Γ Δt. This is the Poisson distribution.

The law of exponential decay

Consider a single unstable particle.
For an infinitesimal time interval dt, the probability of decay is dP = Γ dt.
So properly, Γ is defined as –(Subscript[dP, e])/dt = the limit of –(Subscript[ΔP, e])/Δt as Δt ⟶ 0, where Pe(t) is the probability that the particle exists at time t.

Now consider N(t) = the number of unstable particles in a macroscopic sample at time t.

N(t+dt) = N(t) – N(t) Γ dt

The solution of this differential equation is 

	N(t) = N(0) exp( –Γ t )		⟹ 		exponential decay

Derivation of the Poisson probability distribution

Let P( n ; t ) = the probability that n decays will occur in the time interval t.

Now consider P ( n+1 ; t ).

In order to have n+1 decays in time t, there must be ...

(i) n decays in time t', for some time t’ with t' < t ;
this has probability P(n ;t' ).

(ii) one decay during the infinitesimal time interval from t' to t' + dt' ; this has probability Γ dt' .

(iii) no decays in the interval from t' +dt'  to t;
this has probability P(0 ; t – t' – dt') which we may approximate by
P(0; t – t') treating dt' as infinitesimal.

Now, P( n+1; t) is the product of the three probabilities, integrated over all possible values of t', i.e., from 0 to t :

	P( n+1 ; t ) = ((∫_0)^t)  P( n  ;t' ) Γ dt' P( 0 ; t – t' ).

Now, we know that

	 P( 0 ; t – t' ) = exp{ – Γ (t – t' ) }
	 
from the law of exponential decay, so

	P( n+1 ; t ) = ((∫_0)^t)  P( n  ;t' ) Γ exp{ – Γ (t – t' ) }  dt' .

Now it is straightforward to show that the solution of this recursion relation is 

	P( n ; t ) = (μ^(n )) exp{–μ} / n!	where μ = Γ t
	
which is the Poisson distribution.

An example of a Poisson process is the sequence of radioactive decays in a sample of an unstable nuclear isotope. For example, consider a sample of Ra-226. This isotope is unstable—it undergoes alpha-decay—and the half-life is (Subscript[t, 1/2]) = 1600 years.

The probability per unit time for any single nucleus to decay, which is called the "decay rate", is   Γ = ln(2) / (Subscript[t, 1/2])  = xxx (sec^(-1)).
So if we use the second as a standard time interval, Δt = 1 sec, then the probability of decay during one second is p = xxx << 1. Since this is very small, the process of radioactive decays is a Poisson process.

Note that only the mean μ appears in P(n;t) so that knowledge of N and p is not always necessary. The mean in refers to the mean number in units of the time interval t or Δt.

I.e., µ = Γ Δt, and we can rewrite the Poisson distribution as

P( n ; Δt ) = (Γ ((Δt))^n) exp(–Γ Δt) /n!

EXERCISE 2
P(n) = (μ^(n )) exp(–μ) / n! 
(A) Calculate the theoretical mean of n.
(B) Also calculate the theoretical variance.

EXERCISE 3
Plot the Poisson distribution P(n) for μ = 5, 10, 40.
Put all three plots on one graph.
Remember, the domain of the Poisson distribution is Z ( = the non-negative integers.) So use DiscretePlot.

We see from Exercise 3 that as µ becomes large, the distribution becomes more and more symmetric and approaches a Gaussian form. We are replacing a discrete distribution by a continuous one. This brings us to the third important probability distribution—the Gaussian distribution.

2.3 The Gaussian or Normal Distribution

The Gaussian or normal distribution is the most familiar distribution in all the sciences. Measurement errors, such as instrumental errors, are generally described by the Gaussian distribution. And even if the Gaussian distribution is not strictly correct, it often provides a good approximation to the true governing distribution.

The Gaussian is a continuous, symmetric distribution whose density is given by

EXERCISE 4

(A) Calculate the mean and variance for the Gaussian distribution with parameters μ and σ.

(B) Plot the Gaussian distribution for μ = 0, 1, 2 and σ = 1, 2, 3.

(C) The full width at half maximum (FWHM) for the Gaussian distribution is
FWHM = A σ, where A is a numerical constant. Determine A.

The areas under the Gaussian between integral intervals of sigma are important in the publication and interpretation of measurement errors. For example, if an experimental result is presented as x = a ± b, it means that the probability that the true results lies between a–b and a+b is 68%.

EXERCISE 5

(A) What is the area under the Gaussian distribution, for x ∈ ( μ–σ , μ+σ ).
This is the probability for x to lie within 1-sigma of the mean.

(B) Repeat the calculation for the 2-sigma range.

(C) Repeat the calculation for the 3-sigma range.

In high-energy physics, a fundamental discovery must pass the 5-sigma test.

Radium

All isotopes of radium have half - lives much shorter than the age of the Earth, so that any primordial radium would have decayed long ago.Radium nevertheless still occurs in the environment, as the isotopes 223 Ra, 224 Ra, 226 Ra, and 228 Ra are part of the decay chains of natural thorium and uranium isotopes; since thorium and uranium have very long half - lives, these daughters are continually being regenerated by their decay.[9] Of these four isotopes, the longest - lived is 226 Ra (half - life 1600 years), a decay product of natural uranium.Because of its relative longevity, 226 Ra is the most common isotope of the element, making up about one part per trillion of the Earth' s crust; essentially all natural radium is 226 Ra.[21] Thus, radium is found in tiny quantities in the uranium ore uraninite and various other uranium minerals, and in even tinier quantities in thorium minerals.One ton of pitchblende typically yields about one seventh of a gram of radium.[22] One kilogram of the Earth' s crust contains about 900 picograms of radium, and one liter of sea water contains about 89 femtograms of radium.[23]

The common historical unit for radioactivity, the curie, is based on the radioactivity of 226 Ra.[32]

Some of the biological effects of radium were apparent from the start. The first case of so-called “radium-dermatitis” was reported in 1900, only 2 years after the element’s discovery. The French physicist Antoine Becquerel carried a small ampoule of radium in his waistcoat pocket for 6 hours and reported that his skin became ulcerated. Pierre and Marie Curie were so intrigued by radiation that they sacrificed their own health to learn more about it. Pierre Curie attached a tube filled with radium to his arm for ten hours, which resulted in the appearance of a skin lesion, suggesting the use of radium to attack cancerous tissue as it had attacked healthy tissue.[71] Handling of radium has been blamed for Marie Curie’s death due to aplastic anemia. A significant amount of radium’s danger comes from its daughter radon: being a gas, it can enter the body far more readily than can its parent radium.[35]

Today, 226Ra is considered to be the most toxic of the quantity radioelements, and it must be handled in tight glove boxes with significant airstream circulation that is then treated to avoid escape of its daughter 222Rn to the environment. Old ampoules containing radium solutions must be opened with care because radiolytic decomposition of water can produce an overpressure of hydrogen and oxygen gas.[20]

●

The most common isotope of radium is 88(Ra^226) .
It is unstable and decays to a radon isotope by alpha-decay,
	(Ra^(226 ))⟶ (Rn^222) + α ; half life = 1600 years.
Radon is also unstable, so the decay of radium ultimately produces ionizing radiation,
which can excite fluorescent materials ( radio-luminescence ). In the early 20th century, watch dials were painted with radium to make them luminous. However, radium is toxic,
so today radium has no practical applications outside of nuclear medicine.

In nature, radium is found in uranium ores in trace amounts—about 1/7 gram per ton.

EXERCISE 8
Consider a sample of pure Ra-226, amounting to 1.0(x10^(-6)) gram.
Let Γ = the decay rate for Ra ⟶ Rn + α.
The probability that a radium atom will decay during the infinitesimal time dt is Γ dt.
(A) From the half-life (1600 years) calculate Γ , and express Γ in (((sec))^(-1)) .
(B) Calculate the mean number of decays in 1 hour, and call this number A.
(C) Calculate the probability that the number of decays in 1 hour will be less than 0.25A.
(D) Calculate the probability that the number of decayse in 1 hour will be more than 0.75A.
(E) Plot a graph that shows the probability distribution for the value of A.