Basic concepts of probability theory Random variable discrete/continuous random variable Transform Z transform, Laplace transform Distribution Geometric, mixed-geometric, Binomial, Poisson, exponential, Erlang, Hyper-exponential, phasetype, Markovian arrival process 1
Random variable X is denoted as random variable Discrete random variable, if X is discrete Continuous random variable, if X is continuous Distribution function, or called cdf, cumulative distribution function F(x)=Pr(X<x) Probability density/mass function f(x)=pr(x=x), x is discrete (probability mass function, pmf) f(x)= F(x)/ x, x is continuous (pdf) 2
Random variable Mean: E(X) Variance: Var(X), or σ 2 (X) σ 2 (X) = E{(X-E(X)) 2 }=E(X 2 )-E 2 (X) Standard deviation: σ(x) Covariance of two random variables X, Y Cov(X,Y)=E{(X-E(X))(Y-E(Y))} Correlation coefficient of X, Y r(x,y)=cov(x,y)/ σ(x)σ(y) -1 r(x,y) 1 3
Coefficient of variation: Coefficient of variation: c X c X = σ(x)/e(x) c X =0: deterministic c X <1: smooth c X =1: pure random c X >1: bursty Figures as example more than variance t t t 4
Discrete Random Variables Probability mass function(pmf), P(n) Pn ( ) = Pr[ X= n] wh ere 0 Pn ( ) 1, Pn ( ) = 1 Cumulative distribution function(cdf), F(x) Fx ( ) = Pr[ X x] = Pn ( ) n x where F(0) = 0 F( ) = 1 Fb ( ) Fa ( ), if b a Fb ( ) Fa ( ) = Pr[ a< X< b] n Suppose the random variable X is positive 5
Discrete Random Variables Mean and Variance Mean: x = np(n) n Variance: 2 2 σ = ( n x) P(n) n 6
Continuous Random Variables Probability density function(pdf), f(x) f( x) 0 Pr[ a X b] = f ( x) dx 0 f ( x) dx = 1 b a Cumulative distribution function(cdf), F(x) F( x) = Pr[ X x] = f ( y) dy df( x) f( x) = dx 0 x 7
Continuous Random Variables Mean and Variance Mean: x = xf ( x) dx = xdf( x) 0 0 x = (1 F ( x )) dx 0 Variance: σ 2 2 2 2 ( x x) f ( x) dx x f ( x) dx x = = 0 0 8
Z-transform for discrete distribution Z-transform is also called generating function P(z): Z-transform of discrete r.v. X, p(n)=pr(x=n), assume n=0,1,2, Pz ( ) = Ez ( X ) = pnz ( ) Property n= 0 P(0) = p(0), P(1) = 1, P (1) = EX ( ) n P (1) =? 9
Laplace-transform for continuous distribution F*(s): Laplace transform of a continuous r.v. X pdf of X is f(x), cdf of X is F(x), assume x 0 * sx sx F s = e f x dx = e df x 0 0 ( ) ( ) ( ) Property Shortcut to calculate the k-moment of X * * *( k) k k F(0) = 1, F (0) = EX ( ), F (0) = ( 1) EX ( ) * sx F () s = s e F() x dx 0 10
Geometric distribution p: success probability in a Bernoulli trial X: the number of Bernoulli trials needed to first get one success g(k;p)=pr{x=k}=p(1-p) k-1, k=1,2, Property E(X)=1/p Var(X)=(1-p)/p 2 c X =(1-p) ½ Curve of probability mass function Discrete version of exponential distribution 11
Mixed Geometric distribution n set of independent Bernoulli trials, with success probability p i, respectively, i=1,2,,n Mixed probability is θ i, i=1,2,,n X: the number of Bernoulli trials first get one success of any one from the n set trials n k pmf: gb( k; θ, p) = θipi(1 pi) EX ( ) = θi / n i= 1 p i i= 1 Discrete version of hyper-exponential distribution 12
Binomial distribution p: success probability in a Bernoulli trial X: the number of successes during n Bernoulli trials Probability function n r Prnp ( ;, ) = Pr[ X= r] = p(1 p) r Property E(X)=np n r 13
Negative binomial distribution Definition (also called Pascal distribution) Bernoulli trial with success prob. p, predefined number k of failures has occurred, stop the random number of successes, X, obeys NB distr. k+ r 1 r nb( rk ;, p) = Pr[ X= r] = p(1 p) r Not required, for your reference k 14
Poisson distribution A Poisson random variable X with parameter λ has probability distribution n λ λ PX ( = n) = e, n= 0,1, 2,... n! For the Poisson distribution, it holds X = σ ( X) = λ, cx = 2 1 λ 15
Exponential Distribution (with Parameter μ) Pdf and cdf are f( x) = µ e µ x F( x) = 1 e, x 0 1 2 1 σ x x = σ x = c 1 2 x = = µ µ x f(x) µ 1 Pure random F(x) 0 x 0 16 x
Exponential distribution If X 1, X 2,, X n are independent exponential random variables with parameter μ 1, μ 2,, μ n Then, Y = min(x 1, X 2,, X n ) is an exponential random variable with parameter μ = μ 1 + μ 2 + + μ n How about Z = max(x 1, X 2,, X n )? Prove it as homework 17
Exponential distribution has memoryless property PX ( > x+ t) PX ( > x+ t X> t) = PX ( > t) µ ( x+ t) e µ x = = e = PX ( > x) µ t e P( t < X < x+ t X > t) = P( X < x) = F( x) = 1 e µx Memoryless: Future state only depends on the current state, independent of the history 18
Erlang distribution X: Erlang-k random variable, X= X 1 +X 2 + +X k, X i are independent random variables obeying exponential distribution with para. μ Denote this distribution as E k (μ), its pdf is Its cdf is 19
Erlang distribution Mean,variance, squared coefficient of variation Model smooth data traffic barber shop pdf curve is 20
Hyperexponential distribution X is with prob. p i to select exponential r.v. X i with para. μ i, i=1,,k, denote this distribution as H k Its pdf is Its mean is Its coefficient of variation 1 Model bursty data traffic 21
Phase-type distribution (discrete PH type distribution time case) the first passage time to the absorbing state of a discrete time Markov chain Two para., one is transition probability matrix T, the other is the initial state probability α, n phase k cdf:, pmf: Example Fk ( ) = 1 αte k 1 f( k) = αt t 0.3 0.4 0.3 T t P = 0.2 0.6 0.2 = 0 1 0 0 1 α = [0.3,0.7] Te+ t = e 22
Phase-type distribution (discrete time case) Example Geometric distribution, 1 phase 1 p p T t P = = 0 1 α = [1] 0 1 pmf: Mix-Geometric distribution, n phase 0.3 0 0.7 T t P = 0 0.8 0.2 = 0 1 0 0 1 pmf: f( k) = T t = p(1 p) k 1 k 1 α α = [0.5,0.5] f( k) = T t = 0.5 0.7 0.3 + 0.5 0.2 0.8 k 1 k 1 k 1 α 23
PH type distribution (continuous PH type distribution time case) the first passage time to the absorbing state of a continuous time Markov process Two para., one is transition rate matrix T, the other is the initial state probability α, n phase cdf: Example Tx F( x) = 1 αe e 5 4 1 T t B = 2 6 4 = 0 0 0 0 0 α = [0.8,0.2] Te+ t = 0 24
PH type distribution (continuous Example time case) Exponential distribution: F( x) 1 e Tx = α e= 1 e λ Erlang distribution with (m,λ): α = [1,0,0,...,0] Hyper-exponential distribution α = [ θ1, θ2,..., θ n ] x α = 1, T = λ T t B = 0 0 λ λ λ λ T =.... λ λ1 λ 2 T =... λn 25
Coxian distribution A special case of PH type distribution µ 1 p1µ 1 0 0... (1 p1) µ 1 0 µ 2 p2µ 2 0... (1 p2) µ 2 B = 0 0... µ k 1 pk 1µ k 1 (1 pk 1) µ k 1 0 0... 0 µ k µ k 0 0... 0 0 0 α = [1,0,0,...,0] k phase 26
Pareto distribution Pareto(α): a distribution with a power-law tail pdf: cdf: f x x x f x Usually, 0<α<2 α 1 ( ) = α, 1; otherwise, ( ) = 0 α F( x) = 1 x, x 1; otherwise, F( x) = 0 Pareto distribution decays much more slowly than exponential distribution We call it has heavy tail or fat tail Heavy tail is very important for practical data traffic 27
Relation to Central Limit Theorem CLT: X i are i.i.d. r.v. s with mean μ and variance σ 2, define Z n =(X 1 + + X n - nμ)/(σ n), then Z n converges to the standard normal distribution CLT can explain many natural phenomena obey normal distribution Binomial(n,p) is a sum of i.i.d. Bernoulli(p) r.v. s, it converges to a normal distribution when n is large Poisson(λ) can be approximated by a normal distribution with mean λ and variance λ Binomial and Poisson can be viewed as an approximation of discrete version of normal distribution, plot their pdf 28
An explanation to Additive White Gaussian Noise (AWGN) AWGN explanation Transmit a signal in a channel 100 i.i.d. sources make low noise, uniformly distributed among [-1, 1] Noise is additive, if the total amount of noise level is greater than 10 or less than -10, then it corrupts the signal; otherwise, no problem Calculate the probability of signal no corruption Calculation X~Unif(-1,1), E(X)=0, Var(X)=1/3; S 100 ~Norm(0,100/3) P(-10<S100<10)=Φ( 3)- Φ(- 3)=2Φ( 3)-1=2(0.9572)-1 =0.9144 29
Sum of a random number of r.v. 30
A map of the relation of all these distributions Chalk drawing PH type distribution can almost model any distribution Convert stochastic process to Markovian model Widely used in queuing system, reliable system, regenerative system, etc. Discrete version v.s. continuous version Geometric v.s. Exponential Mixed Geo v.s. HyerExp Binomial, Poisson v.s. Normal Erlang (continuous r.v.) Normal(continuous r.v.) 31