EE 505 Introduction. What do we mean by random with respect to variables and signals?

Transcription

1 EE 505 Introduction What do we mean by random with respect to variables and signals? unpredictable from the perspective of the observer information bearing signal (e.g. speech) phenomena that are not under our control (noise, interference, etc.) measurement uncertainty variability of the source (e.g. different speakers) imperfect models Applications with random signals: Signal processing (noise removal, prediction, identification) Communications (compression, signal design, receiver design) Biomedical diagnostics, neuroengineering Transportation planning Electric power system integration of renewables Examples of random signals: coin flips bus arrival times traffic on a highway web page hits per hour speech, image, video, text 1

2 Probability Basics Review Probability space: (Ω, F, P) P (Ω) = 1 P (A) 0 A F P ( i=1a i )= P (A i )ifa i A j = i j i=1 P ( ) =0 P (A) =1 P (Ā) P (A B) =P (A)+P (B) P (A B) 2

3 Example 1: Two Coin flips Example 2: Roulette wheel 3

4 Conditional Probability P (A B) = P (A B) P (B) B Definition: The events {A i } form a partition if i A i = Ω and A i A j = (non-overlapping). For such a partition you have: Total probability: P (B) =P (B Ω) = j P (B A j )= j P (B A j )P (A j ) Bayes Rule: P (A i B) = P (A i B) P (B) = P (B A i)p (A i ) j P (B A j)p (A j ) Independence of Events P (A B) =P (A)P (B) 4

5 Random Variable: X :Ω R Map a random experiment outcome into a real number, or taking a measurement on the outcome of the random experiment. Note that there can be more than one measurement that you might take, depending on the experiment. 5

6 Mapping a random experiment outcome into a real number allows us to characterize the random experiment with Probability distributions: probability mass function (pmf) for discrete RVs, probability density function (pdf) for continuous RVs Cumulative distribution functions: F (x) = P r(x x) Discrete variables: pmf p X (x) x Z p X (x) 0, p X (s) =1 x P X (A) = x A p X (x) x F X (x) = P (k) k= Continuous variables: pdf f X (x) x R f X (x) 0, P X (A) = F X (x) = A x f(v)dv =1 f(v)dv f(v)dv Mixed distributions: f X (x) =f(x)+(1 ) i p i δ(x x i ) 6

7 Expectations: A way to partially characterize random experiments Expectation E(X) = kp(k) E(X) = k= vp(v)dv Discrete X Continuous X General Expectations E(g(X)) = g(k)p (k) E(g(X)) = k= g(v)p(v)dv Discrete X Continuous X Important special cases: moments E(X k ) Mean = 1st moment E(X) Variance = 2nd central moment E[(X E(X)) 2 ]=E(X 2 ) E(X) 2 7

8 Important Random Variables Discrete-Valued X Name Range Parameters pmf p(x) Mean Variance G X (z) =E[z X ] Bernoulli {0, 1} 0 p 1 ( p x ) (1 p) (1 x) p p(1 p) 1 p + pz n Binomial {0,...,n} 0 p 1 p x (1 p) (n x) np np(1 p) (1 p + pz) n Geometric {0, 1,...} 0 <p<1 (1 p) x p 1 p p 1 p p 2 p 1 (1 p)z Poisson {0, 1,...} 0 <λ λ x e λ x! λ λ e λ(z 1) Continuous-Valued X Name Range Parameters pdf f(x) Mean Variance Ψ X (w) =E[e jwx ] (b a) a+b Uniform [a, b] a<b b a 2 Gaussian [, ] µ, σ 2 1 2πσ e (x µ)2 /2σ 2 µ σ 2 e (jwµ σ2 w 2 ) 2 Exponential [0, ] >0 e x 1 1 jw Erlang [0, ] >0,n>0 n x (n 1) e x Gamma [0, ],r > 0 (n 1)! (x) (r 1) e x Γ(r) n r 2 n 2 r 2 e jwb e jwa jw(b a) n ( jw) n ( jw) r Laplacian [, ] >0 2 e x w Rayleigh [0, ] 2 x e 2 x2 /2 2 (2 π π/2 2 )2 not incl. 8

9 Which distribution would you use? A modem transmits over an error-prone channel, repeating every 0 or 1 bit transmission five times. The channel changes an input bit to its complement with probability p = 0.1, independently for each bit. A modem receiver takes a majority vote of the five received bits to estimate the signal. Heat must be removed from a system according to how fast it is generated. Suppose the system has eight components, each of which is active with probability 0.25, independent of the others. Events are the number of systems that are active. A kid is sitting on the corner watching cars go by, waiting to see a Tesla. The probability that a driver in Seattle owns a Tesla is p =.005. X is the number of cars that he sees before a Tesla drives by. 9

10 EE 505 Class Assignment 1 1. Two transmitters send messages through bursts of radio signals to an antenna. During each time slot each transmitter sends a message with probability 1/3. Simultaneous transmissions result in loss of the messages. Let X be the number of time slots until the first message gets through. What type of random variable would characterize this problem? Specify the sample space and the parameter(s). 2. A data center has 10,000 disk drives. A disk drive fails on a given day with probability What distribution would you use to determine the number of spares to keep on hand? What other information would you need to solve the problem? 1

11 Another useful case: the Zipf RV S X = {1, 2,...,L} P (k) 1 k 10