Random process: A stochastic process or a random process is a family (collection / ensemble) of random variables {X(s,t): s ∈ S, t∈T} defined on the given probability space, indexed by the parameter t where t varies over an indexed set T.

The set of all possible values of {X(s,t): s ∈ S, t∈T} is called its state space and the values of {X(s,t)} are called states.

The parameter t is usually interpreted as ‘time’, even though it may also represent distance, length, thickness etc.

The state space is discrete, if it contains a finite or countable number of points, we have discrete state process or a chain. If the state space is continuous, we have a continuous-stare process.

If the index set T is discrete, we have a discrete (time) parameter process. Otherwise, we have a continuous parameter process.

Notation: For discrete time, the stochastic process is represented by {$X_n$,n∈ N} or simply {$X_n$}. For continuous time, the stochastic process is represented by {X(t) : t∈ T} or simply {X(t)}.

Example: Let the random variable $X_n$ denote the total number of sixes appearing in the first n throws of a die.

Then {$X_n$} is a random process with S= {0,1,2,...,n} (discrete) and T= {1,2,3,...} (discrete).

1. Mean: Mean of the process {X(t)} is the expected value of a typical member X(t) of the process. Mean is denoted by μ (t).

   That is, μ(t) = E(X(t)) = $∫^\infty_{-\infty} xf (x;t ) dx$

2. Autocorrelation: Autocorrelation of the process {X(t)} is the expected value of the product of any two members X($t_1$) and X($t_2$) of the process. Autocorrelation is denoted by

R($t_1$,$t_2$) [ or $R_{XX}$ ($t_1$,$t_2$) or $R_X$ ($t_1$,$t_2$) ]

That is, R($t_1$,$t_2$) = E[X($t_1$) X($t_2$)]

1. Autocovariance: Autocovariance of the process {X(t)} (Denoted by C($t_1$,$t_2$) [ or $C_{XX}$($t_1$,$t_2$) or $C_X$($t_1$,$t_2$)] ) is defined as

C($t_1$,$t_2$) = E[(X($t_1$)-μ($t_1$))(X($t_2$)-μ($t_2$))]

Here, μ($t_1$)= E(X($t_1$) & μ($t_2$)= EX($t_2$)

Note:

C($t_1$,$t_2$) = E[(X($t_1$)-μ($t_1$))(X($t_2$)-μ($t_2$))]

= E(X($t_1$) & μ($t_2$)= EX($t_2$)

i.e. C($t_1$,$t_2$) = R($t_1$,$t_2$) - μ($t_1$)μ($t_2$)

1. Correlation coefficient: Correlation coefficient of the process {X(t)} (Denoted by

p($t_1$,$t_2$) [ or $p_{XX}($t_1$,$t_2$) or $p_X$$t_1$,$t_2$) is defined as p($t_1$,$t_2$) = C($t_1$,$t_2$) / √ C($t_1$,$t_1$)√ C($t_2$,$t_2$) )

Where

C($t_1$,$t_1$)= E[(X($t_1$)-μ($t_1$))(X($t_2$)-μ($t_1$))]

= $E[(X({t_1})-μ({t_1})]^2$

is the variance of X($t_1$) and C($t_2$,$t_2$) is the variance of X($t_2$) .