Probability Theory-Simulations in Statistical Physics-Lecture Slides, Slides of Statistical Physics

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Chapter 2

Topic: Probability Theory

Simulations in

Statistical Physics

Probability Theory

Random Events Monte Carlo calculations are numerical stochastic process; a sequence of random events. Elementary Events: Those events that we can not further analyze into simpler events. For example, results (HEAD or TAIL) of a coin flipping are elementary events. Compound or Composite Events: Those events that are defined from a number of elementary events. For example, results of flipping two coins.

Random Events occur in nature. For example,

• Scattering of electron from an atom,
• Radioactive decay of an unstable nucleus,
• Percolation,
• Scattering of neutrons from solids.

Given an elementary event with a countable set of random outcomes,

E 1 , E 2 , E 3 ,  En

0  pk  1

Another notation for the probability of event Ek is:

There is associated with each possible outcome Ek a number called probability, pk such that :

If the kth outcome never occurs, then probability is zero. pk ^0.

If the kth outcome is sure to occur, then probability is one. pk ^1.

P (Ek )  pk

A property: P{ Ek and /or Ej} pk  pj

Random Events

If two events Ek and Ej are mutually exclusive:

P{ Ek and Ej} 0

P{ Ek or Ej} pk  pj

3. When a whole class of events can be mutually exclusive for all i and j then all the possible events (exhaustive set) will have following property:

 ^1. j

pj

4. A compound event for two elementary events will have probability function of both the probabilities of first and second events. It is called joint probability.

Two events Ei and Ej are mutually exclusive events if and only if the

occurance of Ei implies that Ej does not occur.

Probability

When the occurrence of one event does not affect whether or not some other event happens, then these types of events are called independent.

If events are independent, then knowing that one event has occurred does not change our degree of belief that the other event occur. Two events are said to be independent if and only if any of the following is true:

( ) ( | ).

( ) ( ) ( ), P E P E F

P E F P E P F 

 

This can be extended to k events as

 

    

k

i

P E E E Ek P Ei 1

( 1 2 3  ) ( )

Independent Events

Bayes Theorem

Sometimes we start an analysis with an initial degree of belief that an event will occur. Later on we obtain some additional information about the event that would change our belief about the probability that the event will occur. The initial probability is called prior probability. When we use new information and update our prior probability using Bayes’ theorem we obtain the posterior probability. Consider piston rings are purchased from two manufacturers; 60% from A and 40% from B. If we select a part at random from supply we will have following probabilities:

( ) 0. 4

( ) 0. 6 

 P B

P A

These are prior probabilities that piston rings are from A and B.

Now we write posterior probability as

( ) ( | ) ( ) ( | )

( ) ( | ) ( | ) P A P F A P B P F B

P A P F A P A F 



Note that we have to find two probabilities P(F|A) and P(F|B). These are probabilities that a piston ring will fail given that it came from the corresponding manufacturer.

This is theorem when the outcomes are only two. In general Bayes theorem for n number of mutually exclusive events can be written as

( ) ( | ) ( ) ( | )

( ) ( | ) ( | ) i i n n

i i i P E P F E P E P F E

P E P F E P E F  

 

Bayes Theorem

The expectation of a random variable x is the stochastic mean value and is defined as E (x):

Expected Value

j j

E (x)pj x

It is common to write this as < x >. Consider a real-valued function: g^ (xi ) gi

Where xi are countable elementary events with probability pi. If xi

is a random variable then G(xi) is also a random variable.

The expected value of g(x) is

( ( )) ( ) ( j) j

E g x g x  pj g x

Expectation and nth Moment

From the definition of expected value of a function, < constant > = constant For two functions g and h with two constants, < ag(x) + bh(x) > = a<g(x)> + b<h(x)>

For a linear function g(x): < g(x) > = g()

The nth moment of x is defined as n j j

j x n^ p x  

j j

The First moment of x is  x  pj x

2 2 j j

The second moment of x is x pj x