Stochastic Processes

Stochastic Variables - Home

A random variable is a mathematical object with two properties:

  1. it has a set of possible values called range, set of states or sample space. This set can be continuous or discrete, and it can be one dimensional or multidimensional.
  2. it has a probability distribution associated with this set. The probability distribution function assigns a positive, less than unity number to each element in the set of possible values.

The probability distribution function is normalized, i.e. given the set D of all possible values of random variable X′ then:

∫ DP(x′)dx′ = 1.

When the set of values is discrete then one can talk about the probability of random variable X attaining value x, P(X = x). On a continuous set of values, one cannot talk about the probability of a stochastic variable being a precise value, but rather one has to look at the probability that the random variable is between x and x + dx, P(x)dx

When random variable X has a continuous range, it’s probability distribution functions can contain delta functions.

The cumulative distribution function is ℙ(x):

ℙ(x) = ∫ −∞x+0P(x′)dx′

The expectation value of a function of random variable with range D is defined as:

f(X) = ∫ Df(x)P(x)dx.

When the function is the value of the variable itself, we get the average of the random variable, or it’s first moment:

X = ∫ DxP(x)dx.

The moments of a random variable X with probability distribution function P(x) are:

μm = Xm = ∫ DxmP(x)dx.

The characteristic function of a random variable X is defined by:

G(s) = eisX = ∫ DeisxP(x)dx

.

Rules for addition of scalar random variables Y = X1 + X2:

  1. The average of the sum is the sum of averages. Y = X1 + X2

  2. If variables are uncorrelated then variance of sum is sum of variances. σY 2 = σ X12 + σ X22

  3. If variables are independent , the characteristic function of the sum is the product of characteristic functions. GY (s) = GX1(s)GX2(s)