Note for Communication Engineering - CE by Abhishek Singh

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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

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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)

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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)

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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