what is Event in probability , formulas , Experiment , Sample Space :-

**PROBABILITY THEORY AND RANDOM VARIABLES**

__Inside this Chapter __

- Introduction
- Basic Definitions Related to Probability
- Probability
- Properties of Probability
- Conditional Probability
- Probability of Statistically Independent Events
- Random Variables
- Probability Function or Probability Distribution of a Discrete Random Variable
- Cumulative Distribution Function
- Probability Density Function (PDF)
- Joint Cumulative Distribution Function
- The Joint Probability Density Function
- Marginal Densities
- Conditional Probability Density Function
- Statistical Averages of Random Variables
- Uniform Distribution CI Gaussian or Normal Distribution
- Rayleigh Distribution
- Random Process
- Sum of Random Processes
- Correlation Function
- Spectral Densities
- Response of Linear Systems to Random Inputs

**2.1 INTRODUCTION**

** **In previous semester, in analog communication, we have studied how to apply Fourier transform to analyze some signals. All these signals were described by some fixed mathematical equations. Such type of signals are called **deterministic signals.** The behavior of such type of signals as well as processing through linear time invariant (LTI) systems can be determined with the help of mathematical models. These mathematical models represent the complete behaviour of the signal at every instant of time. Hence, for such deterministic signals, there is no uncertainty about the value at any instant of time. This means that these signals are predictable.

There is one other class of signals, the behaviour of which cannot be predicted. Such type of signals are called random signals. These signals are called **random signals** because the precise value of these signals cannot be predicted in advance before they actually occur. The examples of random signals are the noise interferences in communication systems. This means that the noise interference during transmission is totally unpredictable. In the same way, the noise generated by the receiver itself is random. Even some other signals which are not noise signals are also random signals. These signals cannot be modelled mathematically. Actually the electromagnetic interference is the major source of random noise.

In the receiver, the thermal noise is caused by the random motion of the electrons. Although the random signals are not predictable in advance precisely, they can be described in terms of its statistical properties. It is possible to analyze the random signals statistically with the help of probability theory.

In fact, the probability theory is very essential mathematical tool in the design of digital communication systems.

**2.2 BASIC DEFINITIONS RELATED TO PROBABILITY**

Before we discuss the concept of probability and its applications in the field of communication, we must discuss few basic terms which are related to probability theory.

**2.2.1. Experiment**

An experiment is defined as the process which is conducted to get some results. If the same experiment is performed repeatedly under the same conditions, similar results are expected. But there are few experiments which do

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The theory of probability deals with averages of mass phenomena occurring sequentially or simultaneously; electron emission, telephone calls, radar detection, quality control, system failure, games of chance, statistical mechanics, turbulence, noise, birth and death rates, and queueing theory, among many others. |

not produce the same result as we have stated above. The theory of probability mainly deals with such type of experiments. In fact, the theory of probability is applied to such type of experiments to predict the possibility of a particular output. An experiment is sometimes called **trial**. As an example, throw of a coin is an experiment or trial. This trial results in two outcomes namely **Head** and **Tail**. Similarly, drawing a card from a well shuffled pack is the trial or experiment and it results in 52 possible outcomes (i.e. cards). Hence, each experiment or a trial has an outcome and the possibility of this outcome can be predicted with the help of probability theory.

The outcomes of an event are called **equally** likely, if any one of them cannot be expected to occur in preference to another. As an example, the tossing of a coin results in two outcomes, Head and Tail. Both have same possibility of 50%. Such type of outcomes are called equally likely outcomes. On the otherhand, if the box contains two black and four white balls, then getting a white or black ball is not equally likely. This is possible that we may get white balls sequentially.

**2.2.2. Sample Space**

** **A set of all possible outcomes of an experiment or trial is called the **sample space** of that experiment. It is generally denoted by ‘S”. The total number of outcomes in a sample space is denoted by n(s). As an example, the tossing of a coin has two outcomes. Hence, we may write its sample space as :

S = {H, T}

where, H ” Head

and T ” Tail

Now, we consider another example. If three coins are tossed simultaneously, then each experiment has two outcomes namely H and T. Hence, the maximum possible number of outcomes will be eight.

S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

So that n(s) = 8

** **

**2.2.3. Event**

** **The expected subset of the sample space or happening is called an event. As an example, let us consider an experiment of throwing a cubic die. In this case, the sample space S will be as

S = {1, 2, 3, 4, 5, 6}

Now, if we want the number ‘3’ to be an outcome or an even number, i.e., {2, 4, 6}, then this subset is called an event. This is denoted by letter `E’. Hence

event E is a subset of the sample space ‘S’. If event E has only one outcome, then it is called an elementary event. On the other hand, if event E does not contain any out come, then it is called a null event. If E = S, then an event contains all the outcomes. Such as event is called a **certain event.** It always occurs, no matter what so ever is the outcome.

Two events A and B are called independent events if happening of event A has nothing to do with happening of B. As an example, let us consider an experiment of tossing a coin two times. In this case, occurrence of head in the first throw has nothing to do with the occurrence of head in the second throw. Hence, these two events are independent events since their outcomes are independent. On the other hand, if the outcome of one event is affected by other, then these events are called dependent events.