1. TELECOMMUNICATIONS Dr. Hugh Blanton ENTC 4307/ENTC 5307

2. Random Variables

3. Dr. Blanton - ENTC 4307 - Random Variables 3 Random Variables Many random phenomena have outcomes that are real numbers, e.g., the voltage, v(t) at time, t, across a noisy resistor, number of people on a New York to Chicago train, etc. In engineering, technology, and science; we are generally interested in numerical outcomes. Even when the universal set, S, in not numerical, we may apply a mapping to convert the outcomes to real numbers.

4. Dr. Blanton - ENTC 4307 - Random Variables 4 Definition of a Random Variable: A random variable is a number labeling the outcomes of a probabilistic experiments. X can be considered to be a function that maps all the elements in S into points on the real line or some parts thereof.

5. Dr. Blanton - ENTC 4307 - Random Variables 5 Universal Set, S R X(.) X:S  R (Real numbers) Mapping RangeDomain Conditions: The mapping is single-valued. The set {X  x} is an event. This is the set of random variable X taking values equal or less than x in a trial chance experiment, E.

6. Dr. Blanton - ENTC 4307 - Random Variables 6 Basic Definitions Discrete Random Variable: A random variable that has a countable number of elements in the range. Continuous Random Variable: A random variable that has an uncountably infinite number of elements in the range.

7. Dr. Blanton - ENTC 4307 - Random Variables 7 Random Variables The mapping (function) that assigns a number to each outcome is called a random variable. If the random variable is denoted by X, then the distribution function F(x o ) is defined by

8. Dr. Blanton - ENTC 4307 - Random Variables 8 Example 1: Suppose you match coins with a friend, winning $1 if two coins match and losing $1 if the coins do not match. Example 1:S={HH, HT, TH, TT} s 1 s 2 s 3 s 4 Random Variable: X( s 1 ) = X( s 4 ) = +1 X( s 2 ) = X( s 3 ) = -1 Thus, X1 1 SHHHTTHTT  Single-valued mapping

9. Dr. Blanton - ENTC 4307 - Random Variables 9 In this case, a random variable takes on only a finite number of values (+1, -1), satisfying property c. If we let x = 0.6, then X  0.6, if s = HT or TH, i.e., the event {HT, TH}. Thus x = 0.6 determines an event. Let x = -10, the {X  -10} = Ø Let x > 1, then {X  x} = S Thus, for every x, we have an event and b is satisfied.

10. Dr. Blanton - ENTC 4307 - Random Variables 10 Basic Definitions Discrete Random Variable: A random variable that has a countable number of elements in the range. Continuous Random Variable: A random variable that has an uncountably infinite number of elements in the range. Probability Assignment: There are two standard forms for probability assignment either using Cumulative Distribution Function (CDF) or Probability Distribution Function (PDF).

11. Dr. Blanton - ENTC 4307 - Random Variables 11 Cumulative Distribution Function (CDF) Let X : a random variable with a particular value, x, then, F X (x) = Pr[X  x] Thus, the CDF is the probability of event {X  x}, i.e., the random variable, X, takes on a value equal to or less than x.

12. Dr. Blanton - ENTC 4307 - Random Variables 12 Example 2 Experiment: Observing the parity bit in a word in computer memory. Bit “ON”  X = 1 Bit “OFF”  X = 0 The OFF state has a probability q and thus the ON state has a probability of (1-q). Sample space, S = {OFF, ON} Plot F X (x)

13. Dr. Blanton - ENTC 4307 - Random Variables 13 Example 2 q q F X (x) x Prob. of event {X=0} Prob. of event {X=1}

14. Dr. Blanton - ENTC 4307 - Random Variables 14 Example 3 Determine CDF for a single toss of a die.

15. Dr. Blanton - ENTC 4307 - Random Variables 15 Example 3

16. Dr. Blanton - ENTC 4307 - Random Variables 16 1/6 1 6 F X (x) x 1

17. Dr. Blanton - ENTC 4307 - Random Variables 17 Example 4 A random variable has a PDF given by F X (x) = 0 -  < x  0 = 1-e -2x 0 < x   Find the probability that X > 0.5. Find the probability that X  0.25 Find the probability that 0.3  X  0.7

18. Dr. Blanton - ENTC 4307 - Random Variables 18 Example 4

19. Dr. Blanton - ENTC 4307 - Random Variables 19 F X (x) x 1

20. Dr. Blanton - ENTC 4307 - Random Variables 20 Example 5 A random variable has PDF given by: F X (x) = A(1-e -(x-1) )1< x <  = 0-  < x  1 Find A for a valid CDF FX(x) = ? Pr[2 < X <  ] = ? Pr[1 < X  3] = ?

21. Dr. Blanton - ENTC 4307 - Random Variables 21 Example 5 (a) Since F X (  ) = 1,  A [1 – e -  ]  A = 1 (b) F X (2) = [1 – e -1 ] = 0.6321 Pr[2 < X <  ] = F X (  ) - F X (2) = 1 - 0.6321 = 0.3679 (c) Pr[1 < X  3 ] = F X (3) - F X (1) = (1 – e -2 ) - (1 – e 0 ) = 0.8647

22. Dr. Blanton - ENTC 4307 - Random Variables 22 CDF or Discrete Random Variable: A discrete random variable, X, taking on one of the countable set of possible values x 1, x 2,  with probability Pr[X = x k ],  k  [1,N] forming a stair-step CDF with amplitude of each step being Pr[X = xk], k = 1, 2, . Thus, where, Or more compactly,

23. Dr. Blanton - ENTC 4307 - Random Variables 23 Example 6 A bus arrives at random in (0, T], i.e., 0 < t  T. Let X be a random variable representing time of arrival, then clearly, F X (t) = 0for t  0impossible event F X (T) = 1certain event Bus is uniformly likely to come at any time within (0,T]. Then A continuous random variable has a continuous CDF. 1 0Tt F X (t)

24. Dr. Blanton - ENTC 4307 - Random Variables 24 A PDF is defined as Properties of PDF: If fX(x) exists, then (1) i.e., CDF (2) Probability Density Function (PDF)

25. Dr. Blanton - ENTC 4307 - Random Variables 25 (3) If a = -  and b = , then (4)since CDF is non- decreasing From (2), the probability that X takes on values between x and x +  x is

26. Dr. Blanton - ENTC 4307 - Random Variables 26

27. Dr. Blanton - ENTC 4307 - Random Variables 27 Generalization For discrete random variables, the PDF has a general form of Example 8: For a random variable, X, we have (a)Find A so that this function is a valid PDF. (b) Find Pr[1/2  x 1].

28. Dr. Blanton - ENTC 4307 - Random Variables 28 Example 8 (a)

29. Dr. Blanton - ENTC 4307 - Random Variables 29 Example 8 (cont.) (b)