Classification of Stochastic Process Lecture 11

Classification of Stochastic Process 

Lecture 11 

The collection of all possible values of Xt (discrete or continuous) at any given time t (discrete or continuous) is represented by S or X in a stochastic process, whereas the set of time points (discrete or continuous) is known as the time index. S and T are used to categorise the stochastic process. There are four different types of processes as a result.

1. Discrete Random Sequence (TD, SD)

2. Continuous Random Sequence (TD, Sc)

3. Discrete Random Process (Tc, SD)

4. Continuous Random Process (Tc, Sc)

Discrete Discrete Stochastic Process

(Discrete Stochastic Sequence)

If the random variable "Xt" is discrete and both the time index "T" and state space "S" are discrete. The stochastic process is said to be a discrete stochastic process.

Diagrammatically it can be represented as:

Example: The number of customers (Xt) in the cash counter of a bank at any time t (in hours) of the nth day of the week. The state space of the stochastic process {Xt: t ≥ 9.00}. Here T = 9.00, 10.00, ..., 04.00

Here Xt (number of customers) is discrete, the time index (T) from 09.00 am to 04.00 pm is discrete, and there are five working days in a week, so S is also discrete.

Discrete Continuous Stochastic Process

(Discrete Stochastic Processes)

The stochastic process will be discrete continuous random process, if the random variable (Xt) is continuous and both the time index (T) and state space (S) are discrete.

Diagrammatically represented of Discrete Continuous Stochastic Processes:

Examples: The maximum temperature (Xt) of a city recorded at 2.00 pm on n^th day, if the temperature of the city concerned lies between 25°C and 43°C. The state space of the stochastic process {Xt: t ≥ 1}. Here T = {1, 2, 3, ...} & S = {25°C, 43°C}.

Continuous Discrete Stochastic Process

(Continuous Random Sequence)

The stochastic process will be a continuous discrete random process if the random variable Xt is discrete, the time index (T) is continuous and the state space (S) is discrete.

Visual representation of a continuous discrete random process:

Example: The number of vehicles passing through the toll plaza at the nth time of the n^th day of a week. The state space of the stochastic process {Xt: t ≥ 0}. Here T = {07, 05} and S = {1, 2, 3, 4, 5, 6, 7}.

Continuous Continuous Stochastic Process

(Continuous Random Process)

The stochastic process will be a continuous stochastic process if the random variable is continuous and both the time index (T) and the state space (S) are continuous.

Diagrammatically it can be represented as:

Example: The blood pressure of a patient on the n^th hour of a patient whose blood pressure oscillates between 120/90 and 145/100. The state space of the stochastic process {Xt: t ≥ 0}. Here T = {01, 24} and S = {120/90, 145/100}.

• Read More: Stationary Stochastic Process