Statistical Continuum Mechanics Distribution Functions

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Statistical Continuum MechanicsDistribution
Functions

• H. Garmestani
• Ref Probability, Random VariablesBy Peyton Z.
  Peebles
• Statistical Continuum Theories, Mark Beran,
  volume 7.
• Outline
• Random Variables
• Distribution functions
• Correlation Function
• Distribution functions for one point functions

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Random variable

• Define a random variable (W, X, Y, Z) as a real
  function of elements of a sample space S.
• Use Capital letters (W, X1, X2, X3,) for random
  variables
• and x1,x2,x3, w, x, y for particular values of
  the random variable
• Random variable should be single valued such that
  every point in S, should correspond to only one
  value of the random variable
• Random values can be discrete or continuous

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Random Elementary Event

• Consider here that the random elementary event X
  may take any real number

• For example consider one of the components of the
  the deformation gradients at a random point,x in
  a heterogeneous medium

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Conditions on Random variable

• Two conditions must be satisfied
• The set Xltx shall be an event for any real
  number x
• This corresponds to those points s in the sample
  space for which the random variable X(s) does not
  exceed the number x.
• PXltx is equal to the sum of the probabilities
  of all the elementary events in Xltx
• The second is

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Distribution Function

• The probability PXltx is the probability of
  occurrence of the event X and is a number that
  depends on x,
• FX(x) is the cumulative probability distribution
  function of the random variable X, FX(x)
• Or the distribution function of X. Then
• FX(x) PXltx

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Distribution Function

• Properties of FX(x)

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Distribution Function

• This states that FX(x) is a nondecreasing
  function of x.

• This property is justified for the fact that
  events events Xltx1 and x1 ltXltx2 are mutually
  exclusive so the probability of the events is the
  difference

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Distribution Function

• If X is a discrete random variable, consideration
  of its distribution function
• FX(x) PXltx
• Shows that Fx(x) must be a stairstep function

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Distribution Function (Example)

• Lets assume that in a microstructure, the grain
  misorientation can be represented by one
  misorientation angle (F)! Measurement of the
  number of grains in each misrientation angle
  range results in the following table!

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Distribution Function (Example)
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Distribution Function (Example)

• Now lets find the volume fraction of grains in
  each step!

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Distribution Function (Example)

• Now lets find the distribution Function! The
  distribution Function may be represented as in
  the following.
• Note that the probabilities are then related
  through

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Density Function

• The density function can now be represented as
• Where u() is the unit step function defined as

Use the shortened notation
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Property of Density Functions

• 1-

• 2-

• 3-

• 4-

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Probability Density Function

• Defined as fX(x), is the density function for a
  random variable X, and is the derivative of the
  distribution function

The density function is related to the
probability function through
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Probability Density Function

• Define PX(u(x))du, is the probability that u lies
  between u and udu, or

It is obvious that
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Distribution Function (Example)

• The probability density function for the example
  problem before may be represented as

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Distribution Function (Example)

• Where the distribution Function FX(x) is
  represented as a continuous function with the
  property that

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Binomial Distributions

• Example Lets use the coin toss. The
  probability of obtaining heads is 1/2 as is the
  probability of obtaining tails.

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Binomial Distributions

• Let
• n be the total number of events,
• k be the number of success (or success)
• then the probability of obtaining a particular
  value of k in n total is given by
• B(kn,p)

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Gaussian or Normal Distributions

• The Gaussian Distribution is one of the most
  commonly used and is the limiting value of
  B(xn,p) as Dx0 and n8 for p near 1/2.
• G(x)
• where G(x) is the probability per unit x, µ is
  the limiting value of np, s is the limiting value
  of and

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Gaussian or Normal Distributions

• dPx G(x)dx
• Then P(Xxi) is the probability that a
  measurement lies in the range dx centered at x
• G(x) fX(x)
• Both s and µ are constants.

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Gaussian or Normal Distributions