TELECOMMUNICATIONS Dr. Hugh Blanton ENTC 4307/ENTC 5307

Probability and Random Variables

Dr. Blanton - ENTC Introduction 3 / 30 Probability is the likelihood of the occurrence of a particular event, and is written as P{event}. The probability of an event is a numerical value between zero and unity, where zero implies the event will never occur, and unity implies the event will always occur.

Dr. Blanton - ENTC Introduction 4 / 30 Probability events may include the occurrence of an equality, such as P{x = 5}, or events related to a range of values, such as P{x < 5}.

Dr. Blanton - ENTC Introduction 5 / 30 In contrast to the actual terminology, a random variable is neither random nor a variable, but is a function that maps sample values from a random event or process into real numbers. Random variables may be used for both discrete and continuous processes.

Dr. Blanton - ENTC Introduction 6 / 30 Examples of discrete processes include tossing coins and dice, counting pedestrians crossing a street, and the occurrence of errors in the transmission of data. Continuous random variables can be used for modeling smoothly varying real quantities such as temperature, noise voltage, and received signal amplitude or phase.

Dr. Blanton - ENTC Introduction 7 / 30 Consider a continuous random variable X, representing a random process with real continuous sample values x, where  < x < . Since the random variable X may assume any one of an uncountably infinite number of values, the probability that X is exactly equal to a specific value, x 0, must be zero. Thus, P{X = x 0 } = 0. On the other hand, the probability that X is less than a specific value of x may be greater than zero: 0  P{X  x 0 }  1.

Dr. Blanton - ENTC Introduction 8 / 30 In the limit as x 0 → , P{X < x 0 } → 1, as the event becomes a certainty.

Dr. Blanton - ENTC Introduction 9 / 30 The Cumulative Distribution Function The cumulative distribution function (CDF), F X (x), of the random variable X is defined as the probability that X is less than or equal to a particular value, x. Thus F X (x) = P{X  x}(3.1)

Dr. Blanton - ENTC Introduction 10 / 30 It can be shown that the cumulative distribution function satisfies the following properties: 1.F X (x) > 0(3.2a) 2.F X (  ) = 1(3.2b) 3.F X (-  ) = 0(3.2c) 4.F X (x 1 )  F X (x 2 ) if x 1  x 2 (3.2d)

The Probability Density Function

Dr. Blanton - ENTC Introduction 12 / 30 The probability density function (PDF), f X (x), of a random variable X is defined as the derivative of the CDF:

Dr. Blanton - ENTC Introduction 13 / 30 By the fundamental theorem of calculus, (3.3) can be inverted to give the following useful result: Letting x 1 and x 2 be  and , respectively leads to the fact that the total area under a probability density function is unity.

Some Important Probability Density Functions

Dr. Blanton - ENTC Introduction 15 / 30 There are three basic probability density functions: Gaussian Rayleigh Uniform Poisson The most basic is the PDF of a uniform distribution, defined as a constant over a finite range of the independent variable:

Dr. Blanton - ENTC Introduction 16 / 30 The most basic PDF is the uniform distribution, defined as a constant over a finite range of the independent variable:

Dr. Blanton - ENTC Introduction 17 / 30 Many random variables have gaussian statistics, with the general gaussian PDF given by: where m is the mean of the distribution, and  2 is the variance.

Dr. Blanton - ENTC Introduction 18 / 30 In our study of fading and digital modulation we will encounter the Rayleigh PDF, given by:

Expected Values

Dr. Blanton - ENTC Introduction 20 / 30 Since random variables are nondeterministic, we cannot predict with certainty the value of a particular sample from a random event or process, but instead must rely on statistical averages such as the mean, variance, and standard deviation.

Dr. Blanton - ENTC Introduction 21 / 30 We denote the expected value of the random variable X as or E{X). The expected value is also sometimes called the mean, or average value.

Dr. Blanton - ENTC Introduction 22 / 30 This expected value for a continuous random variables is: