Dept. of Electrical & Computer engineering Probability and Statistics with Reliability, Queuing and Computer Science Applications: Chapter 2 on Discrete Random Variables Dept. of Electrical & Computer engineering Duke University Email: bbm@ee.duke.edu, kst@ee.duke.edu 4/26/2019

Random Variables Sample space is often too large to deal with directly Recall that in the sequence of Bernoulli trials, if we don’t need the detailed information about the actual pattern of 0’s and 1’s but only the number of 0’s and 1’s, we are able to reduce the sample space from size 2n to size (n+1). Such abstractions lead to the notion of a random variable Number : integer or real valued. Examples of discrete RVs are, no. job or pkt arrivals in unit time, no. of rainy days in year etc. Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Discrete Random Variables A random variable (rv) X is a mapping (function) from the sample space S to the set of real numbers If image(X ) finite or countable infinite, X is a discrete rv Inverse image of a real number x is the set of all sample points that are mapped by X into x: It is easy to see that Number : integer or real valued. Examples of discrete RVs are, no. job or pkt arrivals in unit time, no. of rainy days in year etc. Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Probability Mass Function (pmf) Ax : set of all sample points such that, pmf Pmf may also be called discrete density function. Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

pmf Properties Since a discrete rv X takes a finite or a countably infinite set values, the last property above can be restated as, All rv values satisfy p1. Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Distribution Function pmf: defined for a specific rv value, i.e., Probability of a set Cumulative Distribution Function (CDF) Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Distribution Function properties Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Discrete Random Variables Equivalence: Probability mass function Discrete density function (consider integer valued random variable) cdf: pmf: Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Common discrete random variables Constant Uniform Bernoulli Binomial Geometric Poisson Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Constant Random Variable pmf CDF 1.0 c 1.0 c Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Discrete Uniform Distribution Discrete rv X that assumes n discrete value with equal probability 1/n Discrete uniform pmf Discrete uniform distribution function: Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Bernoulli Random Variable RV generated by a single Bernoulli trial that has a binary valued outcome {0,1} Such a binary valued Random variable X is called the indicator or Bernoulli random variable so that Probability mass function: Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Bernoulli Distribution CDF p+q=1 q x 0.0 1.0 Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Binomial Random Variable Binomial rv  a fixed no. n of Bernoulli trials (BTs) RV Yn: no. of successes in n BTs Binomial pmf b(k;n,p) Binomial CDF Notes on pk: pk term signifies n-successes. (cnk : is caused by the fact that there are these many ways in which k 1’s may appear in n-long sequence of 1’s and 0’s e.g. (0,0,1,0,1,1,1,0,0,1,0,0,… 1) Important: each trial is assumed to be an independent trial. Example 2.3 notes: 3-Bernoulli trials has 8-possible outcomes, {000, 001, 010, 100, 011, 101, 110, 111} FX (0)  0 successes: Prob. Of event (000) = 0/125 FX (1)  at least 1 success (i.e. 0 or 1 successes): Prob. Of event (000+001+010+110) = 4x0.125=0.5 FX (2)  at least 2 successes: Prob. of events (000)= FX (1) + Prob(011+101+110) =1-Prob(111)=0.75 FX (3)  1 Symmetric, +ve skewed and –ve skewed Binomial: p=0.5, < 0.5 and > 0.5 As no. of trials n increases (to infinity), B(k;n,p) can be approx. to a normal (Gaussian) distribution Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Binomial Random Variable In fact, the number of successes in n Bernoulli trials can be seen as the sum of the number of successes in each trial: where Xi ’s are independent identically distributed Bernoulli random variables. Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Binomial Random Variable: pmf pk Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Binomial Random Variable: CDF Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Applications of the binomial Reliability of a k out of n system Series system: Parallel system: Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Applications of the binomial Transmitting an LLC frame using MAC blocks p is the prob. of correctly transmitting one block Let pK(k) be the pmf of the rv K that is the number of LLC transmissions required to transmit n MAC blocks correctly; then Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Geometric Distribution Number of trials upto and including the 1st success. In general, S may have countably infinite size Z has image {1,2,3,….}. Because of independence, 1. That is, count the no of trials until the 1st success occurs. Typical output is {00001…} Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Geometric Distribution (contd.) Geometric distribution is the only discrete distribution that exhibits MEMORYLESS property. Future outcomes are independent of the past events. n trials completed with all failures. Y additional trials are performed before success, i.e. Z = n+Y or Y=Z-n The last equation says that, conditioned on Z > n, the no. of trials remaining until 1st success i.e. Y=Z-n has the same pmf as Z had originally. In other words, the system does not remember how many failures it has already encountered. Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Geometric Distribution (contd.) Z rv: total no. of trials upto and including the 1st success. Modified geometric pmf: does not include the successful trial, i.e. Z=X+1. Then X is a modified geometric random variable. Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Applications of the geometric The number of times the following statement is executed: repeat S until B is geometrically distributed assuming …. while B do S is modified geometrically distributed assuming …. Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Negative Binomial Distribution RV Tr: no. of trials until rth success. Image of Tr = {r, r+1, r+2, …}. Define events: A: Tr = n B: Exactly r-1 successes in n-1 trials. C: The nth trial is a success. Clearly, since B and C are mutually independent, We can now also define the modified –ve binomial distribution. The resulting rv is defined as: Just the no. of failures until rth success. Event A gets re-defined as, “Tr = n+r”  Event that there are n failures. Event B: exactly r-1 successes in n+r-1 trials. Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Poisson Random Variable RV such as “no. of arrivals in an interval (0,t)” In a small interval, Δt, prob. of new arrival= λΔt. pmf b(k;n, λt/n); CDF B(k;n, λt/n)= What happens when The problem now become similar to Bernoulli trials and Binomial distribution. Divide the interval [0,t) into n sub-intervals, each of length t/n. For a sufficiently large n, These n intervals can be thought as constituting a sequence of Bernoulli trials, with success probability p= λt/n . So now the problem can again re-defined as, finding the prob of k arrivals in a total of n intervals each of duration t/n. Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Poisson Random Variable (contd.) Poisson Random Variable often occurs in situations, such as, “no. of packets (or calls) arriving in t sec.” or “no. of components failing in t hours” etc. Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Poisson Failure Model Let N(t) be the number of (failure) events that occur in the time interval (0,t). Then a (homogeneous) Poisson model for N(t) assumes: 1. The probability mass function (pmf) of N(t) is: Where l > 0 is the expected number of event occurrences per unit time 2. The number of events in two non-overlapping intervals are mutually independent Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Note: For a fixed t, N(t) is a random variable (in this case a discrete random variable known as the Poisson random variable). The family {N(t), t  0} is a stochastic process, in this case, the homogeneous Poisson process. Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Poisson Failure Model (cont.) The successive interevent times X1, X2, … in a homogenous Poisson model, are mutually independent, and have a common exponential distribution given by: To show this: Thus, the discrete random variable, N(t), with the Poisson distribution, is related to the continuous random variable X1, which has an exponential distribution The mean interevent time is 1/l, which in this case is the mean time to failure Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Poisson Distribution Probability mass function (pmf) (or discrete density function): Distribution function (CDF): Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Poisson pmf pk t=1.0 Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Poisson CDF t 1 2 3 4 5 6 7 8 9 10 0.5 0.1 CDF t=1.0 Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Poisson pmf pk t=4.0 t=4.0 Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Poisson CDF t CDF 1 2 3 4 5 6 7 8 9 10 0.5 0.1 t=4.0 Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Probability Generating Function (PGF) Helps in dealing with operations (e.g. sum) on rv’s Letting, P(X=k)=pk , PGF of X is defined by, One-to-one mapping: pmf (or CDF) PGF See page 98 for PGF of some common pmfs GX(z) is identical to the z-transform (digital filtering) of a discrete time function. If |z| < 1 (i.e. inside a unit circle), the above summation is guaranteed to converge to 1. Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Discrete Random Vectors Examples: Z=X+Y, (X and Y are random execution times) Z = min(X, Y) or Z = max(X1, X2,…,Xk) X:(X1, X2,…,Xk) is a k-dimensional rv defined on S For each sample point s in S, Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Discrete Random Vectors (properties) Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Independent Discrete RVs X and Y are independent iff the joint pmf satisfies: Mutual independence also implies: Pair wise independence vs. set-wide independence Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University

Discrete Convolution Let Z=X+Y . Then, if X and Y are independent, In general, then, Bharat B. Madan, Department of Electrical and Computer Engineering, Duke University