Characteristics of Random Variables

As can be seen from the previous discussion, outcome from various experiments results in different types of random variables. To understand the nature of the outcomes (values assigned to the random variable) from various experiments and to correlate them to the theoretical probability density or probability mass functions it is necessary to understand the characteristics of the random variables.

Similar to the numerical descriptors used in describing the data, random variables can also be understood using some parameters.  Moment about the origin and moment about the mean gives important descriptors of the random variable. Moment about the origin results in mean and the second moment about the mean results in variance.

Moment about the origin

Moments about the origin are called the origin moments. denotes the k^th moment about the origin.

is the the k^th moment about the origin for the discrete random variable.

is the k^th moment about the origin for the continuous random variable.

Moments about the mean

Moments about mean are called central moments and µ_k denotes the k^th moment about the mean.

is the k^th moment about the mean for the discrete random variable.

is the k^th moment about the mean for the continuous random variable.

Expectations

For any random variable X and a function of random variable X, denoted with Y= H(X) (Y is also a random variable) the expected values are denoted with E(X) and E(H(X)), respectively.

If X is a discrete random variable and p(x) is the probability function of X then

           

if X is a continuous random variable and f(x) is the probability density function of  X then

           

The expectation E(X) is called the arithmetic mean or simply the mean of X, and is denoted by µ. The expectation E((X-µ)²), is the variance. Origin and central moments, discussed earlier, also be expressed using the expected value operator.

Origin moments are expressed as follows;

           

Central moments are expressed as follows;

           

When a random variable X is multiplied by a constant the resulting mean and variances are as follows;

E(aX) = a E(X)

V(aX) = a² V(X)

Where, V(X) is the variance of the random variable X.

Expectation of a constant is the constant itself.