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KIIT University
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B.Tech
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Page-1

- Unit 1-Probability, Random Variables, Noise - ( 1 - 9 )
- Unit 2-Signals - ( 10 - 19 )
- Unit 3-Amplitude Modulation - ( 20 - 45 )
- Unit 4-AM Receiver - ( 46 - 50 )
- Unit 5-Frequency Modulation - ( 51 - 64 )
- Unit 5-FM Receiver - ( 65 - 68 )
- Unit 7-Pulse Modulation - ( 69 - 88 )
- Unit 7-Digital Modulation - ( 89 - 104 )
- Unit 9-Communication Systems - ( 105 - 116 )
- List of Formulae from all unit - ( 117 - 132 )

Topic:

EC-3009 :: COMMUNICATION ENGINEERING
(Credit - 4)
Unit One: Introduction
(Probability, Random Variables, Noise)
Jabir Hussain
(jabir.hussainfet@kiit.ac.in)
School of Electronics Engineering
KIIT University
December 8, 2017

Probability
1
Probability
1.1
Definitions
A few basic terms related to probability theory are discussed below.
1. Experiment: An experiment is a set of rules that governs a specific operation that is
being performed. Select at random one card from a deck of 52 cards and write the suit
and number.
2. Trial: A trial is the performance or exercise of that experiment. Someone performs
the experiment.
3. Outcome: An outcome is the result of a given trial. The card selected is 3 of spades.
The outcome of another trial is 10 of clubs.
4. Event: An event is an outcome or any combinations of outcomes. 3 of spades, 10 of
clubs, or any 252 possible combinations.
5. Sample space: A sample space is a set of all possible events.
1.2
Axiomatic Theory of Probability
Axiom 1. The probability of an event E is a non-negative real number.
P (E) ≥ 0
(1)
Axiom 2. The probability of the sample space S is 1.
P (S) = 1
(2)
Axiom 3. The probability of mutually exclusive events E1 , E2 , . . . satisfies
P
∞
[
i=1
!
Ei
=
∞
X
P (Ei )
i=1
Using the axioms, we can define some properties of probability.
1
(3)

Probability
Property 1. The probability of a certain event, E is 1
P (E) = 1
(4)
Property 2. The probability of any event E always satisfies
0 ≤ P (E) ≤ 1
(5)
Property 3. The probability of any two events E1 and E2 is given by
P (E1 + E2 ) = P (E1 ) + P (E2 ) − P (E1 E2 )
(6)
where, P (E1 E2 ) is the joint probability of the events, i.e., the probability
that the events occur simultaneously.
Property 4. Relative Frequency definition which states that the probability of a event
E is given by
P (E) =
Number of favourable outcomes
Total number of outcomes
2
(7)

Random variables
2
Random variables
A random variable, X is a deterministic function that assigns each possible outcome
from the sample space, S of an experiment to a real number.
X:S→R
(8)
A random variable is a function of the outcome of an experiment. Therefore, since each
outcome has a probability, the number assigned to that function by a random variable,
also has a probability. Two ways to analyse experiments via random variables is through
the Cumulative Distribution Function (CDF) and probability density function (pdf).
2.1
Cumulative Distribution Function
Let X be a random variable defined on a sample space S. Then Cumulative Distribution
Function of X, denoted by FX (x0 ), is defined as
FX (x0 ) = P (X ≤ x0 )
(9)
where, P (X ≤ x0 ) is the probability that value of the random variable X is less than or
equal to a real number x0 .
It has the following properties:
Property 1. 0 ≤ FX (x0 ) ≤ 1.
Property 2. FX (+∞) = 1, and FX (−∞) = 0.
Property 3. For x1 ≤ x2 , FX (x1 ) ≤ FX (x2 ).
3

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