Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
In the Name of the Most High Performance Evaluation of Computer Systems Introduction to Probabilities: Discrete Random Variables By Behzad Akbari Tarbiat Modares University Spring 2009 These slides are based on the slides of Prof. K.S. Trivedi (Duke University) 1 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 2 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 3 Probability Mass Function (pmf) Ax : set of all sample points such that, pmf 4 pmf Properties Since a discrete rv X takes a finite or a countable infinite set values, the last property above can be restated as, 5 Distribution Function pmf: defined for a specific rv value, i.e., Probability of a set Cumulative Distribution Function (CDF) 6 Distribution Function properties 7 Discrete Random Variables Equivalence: Probability mass function Discrete density function (consider integer valued random variable) cdf: pmf: pk P( X k ) F ( x) x k 0 pk pk F (k ) F (k 1) 8 Common discrete random variables Constant Uniform Bernoulli Binomial Geometric Poisson 9 Constant Random Variable pmf 1.0 c CDF 1.0 c 10 Discrete Uniform Distribution Discrete rv X that assumes n discrete value with equal probability 1/n Discrete uniform pmf Discrete uniform distribution function: 11 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: p P( X 1) q 1 p P( X 0) 12 Bernoulli Distribution CDF p+q=1 q 0.0 1.0 x 13 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 14 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: Y n X 1 X 2 ... X n where Xi ’s are independent identically distributed Bernoulli random variables. 15 Binomial Random Variable: pmf pk 16 Binomial Random Variable: CDF 1.2 1 CDF 0.8 0.6 0.4 0.2 0 0 1 2 3 4 5 6 7 8 9 10 x 17 Applications of the binomial Reliability of a k out of n system n n j k j k Rkofn 1 B(k 1; n, R) b( j; n, R) ( nj )[ R] j [1 R]n j Series system: n Rseries b(n; n, R) ( )[ R] [1 R] Parallel system: j n n j j n j [ R] n n R parallel 1 b(0; n, R) ( nj )[ R] j [1 R]n j 1 [1 R]n j 1 18 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 p K (1) b(n; n, p) p n p K (2) [1 (1 p) 2 ]n p n and p K (k ) [1 (1 p) k ]n [1 (1 p) k 1 ]n 19 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, 20 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 21 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. 22 Applications of the geometric The number of times the following statement is executed: repeat S until B is geometrically distributed assuming …. The number of times the following statement is executed: while B do S is modified geometrically distributed assuming …. 23 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, 24 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 25 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. 26 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: lt / k !e P N t k k lt k 0, 1, 2, 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 27 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. 28 Poisson Failure Model (cont.) successive interevent times X1, X2, … in a homogenous Poisson model, are mutually independent, and have a common exponential distribution given by: The PX 1 t 1 e lt To show this: t0 P ( random X 1 t ) variable, P ( N ( tN(t), ) 0with ) ethe Poisson the discrete 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 lt Thus, 29 Poisson Distribution Probability mass function (pmf) (or discrete density function): k ( l t ) pk PN (t ) k e lt k! Distribution function (CDF): ( lt ) k F x e k 0 k! x lt 30 Poisson pmf pk lt=1.0 31 Poisson CDF CDF 1 lt=1.0 0.5 0.1 1 2 3 4 5 6 7 8 9 t 10 32 Poisson pmf pk lt=4.0 lt=4.0 33 Poisson CDF CDF 1 lt=4.0 0.5 0.1 1 2 3 4 5 6 7 8 9 10 t 34 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 35 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, 36 Discrete Random Vectors (properties) 37 Independent Discrete RVs X and Y are independent iff the joint pmf satisfies: Mutual independence also implies: Pair wise independence vs. set-wide independence 38 Discrete Convolution Let Z=X+Y . Then, if X and Y are independent, In general, then, 39