ECE Lecture 4. Overview Simulation & MATLAB

Transcription

1 ECE Lecture 4 Overview Simulation & MATLAB Random Variables: Concept and Definition Cumulative Distribution Functions (CDF s) Eamples & Properties Probability Distribution Functions (pdf s) 1

2 Random Number Generators Computers are inherently deterministic, and can t generate random numbers Instead they use various deterministic algorithms to generate pseudo-random numbers Such algorithms, called random number generators (RNG s) start with a seed, that determines the sequence of pseudo-random numbers generated Any time the RNG starts with same seed, the same pseudo-random numbers will result To avoid getting the same sequence on repeated runs, the seed is often a function of the computer clock time. 2

3 Computer Simulation A simulation is a computer program that replicates the behavior of some device or system; or the eecution of some eperiment. Simulation gives us insight into the operation of the actual system (or replicates the eperiment) without requiring us to actually construct the system or eecute the eperiment Hence, simulation saves money! A simulation program you already know: PSPICE Maybe you also know: Simulink 3

4 Using MATLAB to Generate Random Integers [MATLAB Help] randi: Pseudorandom integers from a uniform discrete distribution. R = randi(imax,n) returns an N-by-N matri containing pseudorandom integer values drawn from the discrete uniform distribution on 1:IMAX. randi(imax,m,n) or randi(imax,[m,n]) returns an M-by-N matri. R = randi([imin,imax],...) returns an array containing integer values drawn from the discrete uniform distribution on IMIN:IMAX. Eample: 2 rows, 4 columns of random integers between 0 and 10 (incl.) >> my_rand = randi( [0, 10], [2, 4]) my_rand =

5 Eample with MATLAB m-file Say we want to simulate the toss of 2 dice; let X be the # showing on one die, and let Y be the # showing on the other die; Also say that Z is the sum of the 2 # s showing: Z = X + Y Suppose that we want to eecute the eperiment 1000 times, and find the probability that Z = 7. % program dice X = randi([1, 6], [1, 1000]); Y = randi([1, 6], [1, 1000]); Z = X + Y; flag7 = (Z == 7); count7 = sum(flag7) prob = count7/1000 5

6 Course Overview Where Are We Now? I. Basic Probability Rules, Definitions II. Random Variables III. Random Processes An Introduction 6

7 The Random Variable Concept Say we perform an eperiment (say E), the (numerical) outcome of which we will call X. X is a variable, in the sense that it can take on (possibly) many values; X is random, in the sense that we cannot predict (with any certainty) what the outcome will be, a priori. Random variables allow us to give numerical descriptions of the outcome of an eperiment. A random variable is continuous if it can take any value over a continuous range 7

8 Random Variables: Definition A random variable is a mapping (or a function) the domain is the sample space of an eperiment; the range is some subset of the set of real numbers. Random variables assign a numerical value to every possible eperimental outcome. Eample: say we toss 2 coins, and define a RV which counts how many heads appear: S HH TH HT TT R Outcome TT TH HT HH RV X 8

9 RV s: Another Eample Eperiment: Toss a single die Sample space S = { } (Arbitrarily) Define RV X such that X is five times the # showing on the die (Arbitrarily) Define RV Y such that Y is 3 more than the # showing on the die Die X 5 10 Y 4 5 9

10 RV s: Continued Eample To answer probability questions about the RV, back up, to get an equivalent question about the eperimental outcome Eamples: Pr(X = 10) = Pr(die = 2) = Pr(Y = 8) = Pr(die = 5) = ; Pr(Y = 5.5) = Pr(X > 10) = Pr(die = 3, 4, 5, or 6) = Pr(Y 6) = Pr(Y = 4, 5, or 6) = Pr(die = ) = Die X Y

11 Probability Distribution Function (also called Cumulative Distribution Function, CDF) Defn: The cumulative distribution function CDF for the random variable X is F X () = Pr(X ) (Note: It is a probability.) the RV a real # Eample: F X () Die X F X () -1 Pr(X -1) = 4 Pr(X 4) = 11

12 CDF Eample, continued Die X F X () = Pr(X ) F X () F X ()

13 CDF Properties 1. 0 F X () 1 (because it s a probability) 2. F X (- ) = 0, F X ( ) = 1 3. F X () is monotone non-decreasing 4. Pr(a < X b) = F X (b) F X (a) 5. Typical cdf s a b 1 F X () F X () (for a continuous RV) (for a discrete RV) 13

14 Special Case 1 - The Uniform RV Consider a RV with CDF F X () = 0 a 1 > b linear from 0 to 1 on (a, b) 1 a b This RV is said to be uniformly distributed on (a, b); Notation: U(a, b) Eample of a continuous RV 14

15 Eample: Say X is U(0, 2) 1 F X () 0 2 Some Conclusions Pr(X 1) = F (1) = Pr(X ½) = = Pr(X 1.5) = = Pr(X 27) = = Pr(X > ½) = 1 Pr(X ½) = 1 - F (½) = (corollary 1) Pr(X (0, 1]) = Pr(0 < X 1) = F (1) F (0) = - = Pr(X = 1) = ; Pr(X (0, 1)) = 15

16 Discrete & Continuous RV s Discrete RV s take only a countable # of values, and have stair-case CDF s The probability of the RV taking any specific value is given by the size of the jump in the CDF at that value Continuous RV s take on a continuum of values, and have continuous CDF s The probability of the RV taking any specific value is 0 (also the size of the (non-eistent) jump in the CDF at that value) Uniform RV s are continuous RV s 16

17 Continuous RV, Generic F X () 1 a 0 b General, arbitrary shape Say RV X is continuous on (a, b) Pr(X = 0 ) = 17

18 CDF for Continuous RV An Eample Eperiment: say we are testing diodes, starting at t = 0; Define RV T = time to failure Say F T (t) = Pr(T t) = 0, t < 0 1 ep(-mt), else Pr(diode fails between* times t = a and t = b) m is a parameter of the distribution = Pr(diode fails before t = b) Pr(diode fails before t = a) = F T (b) F T (a) = (1 ep(-mb) (1 ep(-ma) = ep(-ma) ep(-mb) * Don t worry about the endpoints, since the RV is continuous. 18

19 Eample, continued F T (t), with m = For m = 2, assume that t is measured in months; find the probability that the diode failure occurs between 1 and 2 months. Pr(failure between 1 and 2 months) = F T (2) F T (1) = [1 ep(-2(2)] [1 ep(-2(1)] = ep(-2(1)) ep(-2(2)) =

20 More CDF Properties, & Classification Pr(X = a) = D a (size of jump in F X ()) F X () is right-continuous ( holes filled in on top, open on bottom) F X () F X () continuous X is a continuous RV stair-case X is a discrete RV else X is a mied-type RV F X () is monotone non-decreasing; i.e., 1 < 2 F X ( 1 ) F X ( 2 ) 1 20

21 CDF Review 21

22 Probability Density Functions (PDF s) Consider a continuous RV, X Recall F X () Pr(X ), CDF Define f X () df X (), the pdf for the RV X d F X () = f(t) dt ( ] a b Note: Pr(a < X b) = Pr(X b) Pr(X < a) = F X (b) F X (a) = f (t) dt - = b f a (t) dt b a f (t) dt 22

23 PDF s f X () Pr(a < X b) a b Notes f X () 0 for all Pr( < X + d) f ()d f X () d 1 f X () +d d f X () 23

24 CDF, PDF Facts F X () and f X () are each complete descriptions of the RV X Knowing one, we can always find the other Hence they are information equivalent Knowing either one enables us to answer all probability questions 24

25 Eample: Say X is U(2, 5) Then Thus, 1 F X () 2 5 f X (): uniform Slope = rise/run = 1/3 df/d, on (2,5) 1/3 Check: Area = Pr(1 < X < 3) = 25

26 Generalization: PDF s for Discrete RV s Recall for Discrete RV s, F X () is stair-case Eample: 1 F X () d/d f X () ½

27 Dirac Delta Function: Review Dirac delta: d() Area = 1 (shown in parentheses) Amplitude = Shifted Delta: d(-a) Sifting Property of Delta Functions f() d ( a)d Note: In Probability, the area of the delta function at a (in the pdf) is the height of the jump in F X (), or Pr(X = a). 0 (1) 0 a (1) 27

28 Specific Eample: Discrete RV Eperiment: Transmit 3 bits over a noisy channel, where errors occur independently from bit-to-bit, with probability 0.1. If RV X is the # of errors appearing in a 3-bit word on reception, find the pdf and cdf for X. Solution: RV X can take on any of the 3 values: 0, 1, 2, or 3, with the following probabilities: X = k Pr(X = k ) (.1) (.9) (.1) (.9)

29 Eample, continued X = k Pr(X = k ) (.1) 2 (.9) f X () F X () S = 1 (?) 1 (not to scale)

30 Properties of PDF s 1. The area under the entire pdf is 1: 2. To use a pdf to calculate a probability: Pr(a < X b) = 3. PDF s are never negative: f X () 0 4. F X () = f X (t) dt 30