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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.
Probability is a measure of the likelihood that some event will occur. It is always in range of 0 and 1.
A probability distribution of random variable describes the probabilities associated with each possible value for the variable.
If there are fi outcomes out of n for an event i, then probability
p(E) = fi / n.
When we have continuous variable then probability density function is used.
Probability Axioms:
- The probability of an event E must be between 0 and 1.
- Sum of all probabilities for a sample sapce of experiment is one.
- For mutually exclusive events A, B and C, the
- P(A U B U C ) = p(A) + p (B) + p(C)
Probability Theory
If two events Ek and Ej are mutually exclusive:
P{ Ek and Ej} 0
P{ Ek or Ej} pk pj
- 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
- 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
A conditional probability P( E | F ) is the probability that event E is occuring given that the event F has occurred.
A conditional probability is defined for independent events. It is needed when we have partial information concerning the experiment.
; ( ) 0. ( )
( ) ( | )
P F P F
P E F P E F
Here P( E ∩ F ) represents the joint probability that both E and F occur together and P(F) is the probability that event F occurs. We rearrange the equation as
P( EF) P(F)P(E|F).
Conditional Probability
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.
Say we want to know the probability that a piston ring that subsequently failed can from manufacturer A. This would be posterior probability that it came from A and given that the piston ring failed. So, we estimate
; ( ) 0. ( )
( ) ( | ) P F P F
P A F P A F
( )
( ) ( | ) ( )
( ) ( | ) P F
P A P F A P F
P A F P A F
Using multiplication rule we can write numerator in terms of event F and our prior probability that part came from manufacturer A as
Next step is to find P(F). The only way that a piston ring failed is (1)
if it failed and came from A; (2) it failed and came from B
P( F)P(AF)P(BF)
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
Example:
Flipping of a coin has been done. Assign 1 to the event of getting head and 0 to the event of getting tail. Using two different functions g(x) and h(x) let us find the expected values.
Events pi xi g(x) h(x)
HEAD 1/2 1 4 2
TAIL 1/2 0 1 1
g ( x) 1 3 x x
x h x
1
1 3 ( )
x 1 / 2 g( x) 5 / 2 h( x) 3 / 2
Expected Value
