University of Central FloridaSchool of Electrical Engineering and Computer ScienceCOT 4600 - Operating Systems.Fall 2009 - dcmProbability and Statistics ConceptsRandom Variable: a rule that assigns a numerical value to each p ossible outcome of an ex-periment. All possible outcomes of the experiment constitute a sample space.A random variable X on a sample space S is a function X : S 7→ R which assigns a realnumber X(s) to every sample point s ∈ S. This real number is called the probability of thatoutcome.A discrete random variable maps events to values of a countable set e.g., the set of integers);each value in the range has a probability greater than or equal to zero.Example 1. the experiment is a coin toss; the outcome is either 0 (head) or 1 (tail). If thecoin is fair thenp0=p1= 0.5; this means that in a large number of coin tosses we are likelyto observe heads in about half of the cases and tails in the other half of the cases. Anotherexample: when you throw throw a dice the outcome could be 1, 2, 3, 4, 5, or 6; for a fair dicep1= p2= p3= p4= p5= p6= 1/6.A continuous random variable maps events to values of an uncountable set (e.g., the realnumbers).Example 2. the experiment is to measure the speed of cars passing through an intersection:the speed could be any value between 15 and 80 miles/hour. the probability of observing carswith a speed of 19.1 miles/hour could be zero but the probability of observing cars with aspeed from 15 to 19.1 miles/hour could be P19.1= 0.3 which means that 30% of the cars weobserved have a speed in the range we considered.A discrete random variable X has an associated probability density function, (also calledprobability mass function) pX(x) defined as:pX(x) = Prob(X = x)and a probability distribution function also called cumulative distribution function, PX(x)defined as:PX(t) = Prob(X ≤ t) =Xx≤tpX(x)Example 3. You have a binary random variable X (the outcome is either 0 or 1) and:p0= Prob(X = 0) = q and p1= Prob(X = 1) = p, with p + q = 1.Bernouli trials: call the outcome of 1 a “success” and ask the question what is the probabilityYnthat in n Bernoulli trials we have k successes:pk= Prob(Yn= k) =µnk¶pk(1 − p)n−k=n!k!(n − k)!pk(1 − p)n−k1The binomial cumulative distribution function is:B(t : n, p) =tXk=0µnk¶pk(1 − p)n−kExample 4. You have again Bernoulli trials and ask the question how many trails you needbefore the first “success”. If the first success occurs at the i-th trial thenpZ(i) = qi−1pThis is called a geometric distribution. It is easy to prove that:∞Xi=0qi−1p =p1 − q= 1.A continuous random variable X has an associated probability density function, (also calledprobability mass function) pX(x) defined as:fX(x) = Prob(X = x)and a probability distribution function also called cumulative distribution function, FX(x)defined as:FX(t) = Prob(X ≤ t) =Zt−∞fX(x)dxThe expectation of random variable X: E[X] is defined byE[X] =PixipX(xi) if X is discreteR+∞−∞xf(x)dx if X is continuousThe variance Var[X] and standard deviation, σ of random variable X are defined by:Var[X] = σ2=Pi(xi− E[X])2pX(xi) if X is discreteR+∞−∞(x − E[X])2f(x)dx if X is continuousThe moment of order k of random variable X is defined as:E[Xk] =PixkipX(xi) if X is discreteR+∞−∞xkf(x)dx if X is continuousThe centered moment of order k of random variable X is defined as the k-th moment of therandom variable x − E[X]:µk= E£(X − E[X])k¤=Pi(xi− E[X])kpX(xi) if X is discreteR+∞−∞(x − E[X])kf(x)dx if X is continuous2Examples of common distributions1. Uniform distribution in the interval [a,b]: see Figure 1.fX(x) =½1b−aif a ≤ x ≤ b0 if x < a or x > bFX(x) =0 if x < ax−ab−aif a ≤ x ≤ b1 if x > b1a ba b1Figure 1: Probability density function (PDF) and cumulative distribution function of a uni-form distribution2. Standard normal distribution:φ(x) =1√πe−x2/23. Normal distribution with mean µ and standard deviation σ:f(x) =1σ2√πe−(x−µ)2/(2σ2)The probability density function (PDF) and the cumulative distribution function (CDF) of anormal distribution are displayed in Figures 2 and 3.30.80.60.40.20.0−5 − 5313x1.0−1 0 2 4−2−40,μ=0,μ=0,μ=−2,μ=20.2,σ=21.0,σ=25.0,σ=20.5,σ=Figure 2: Probability density function (PDF) of a normal distributionx0.80.60.40.20.01.0−5 − 5313 −1 024−2−40,μ=0,μ=0,μ=−2,μ=20.2,σ=21.0,σ=25.0,σ=20.5,σ=Figure 3: Cumulative distribution function (CDF) of a normal distribution4. Exponential distribution with parameter λ. The probability density function (PDF) f(x)and the cumulative distribution function (CDF), F(x) of an exponential distribution aredisplayed in Figures 4 and 5.4f(x) =½λxe−λxif x ≥ 00 if X < 0F (x) =½1 − e−λxif x ≥ 00 if X < 0Figure 4: Probability density function (PDF) of an exponential distributionFigure 5: Cumulative distribution function (CDF) of an exponential distribution5Elements of Queuing TheoryQueuing theory describ es systems where you have a server and customers and both thearrival of the customers and the time it takes to service them are random variables with acertain distribution. There are two stochastic (aleatory) processes involved:• The arrival process describes the pattern of the customers arrive. The arrival processesis characterized by its probability distribution function e.g., a uniform, exponential,hyper exponential. The distribution can described by one or more parameters such asthe the average value of the random variables subject to that distribution, e.g., thearrival rate denoted as λ or the the inter-arrival time (the average time between twoconsecutive customer arrivals)1λ. For example, if customers arrive in average at twominutes intervals the arrival rate is 1/2 customers/minute and the inter-arrival time is2 minutes.• The service process which describes how customers are served. The service processes ischaracterized by its probability distribution function e.g., a uniform, exponential, hyperexponential. The distribution can described by one or more parameters such as the theaverage value of the random variables subject to that distribution, e.g., the service ratedenoted as µ or the the service time (the average time between two consecutive customerdepartures from the system)1µ. For example, if the service rate is µ = 10 customers perhour, then the service time is 1/µ = 60/10 = 5 minutes.(a) Single