1. 1 Performance Evaluation of Computer Systems By Behzad Akbari Tarbiat Modares University Spring 2009 Introduction to Probabilities: Discrete Random Variables These slides are based on the slides of Prof. K.S. Trivedi (Duke University) In the Name of the Most High

2. 2 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 2 n to size (n+1). Such abstractions lead to the notion of a random variable

3. 3 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

4. 4 Probability Mass Function (pmf) A x : set of all sample points such that, pmf

5. 5 pmf Properties Since a discrete rv X takes a finite or a countable infinite set values, the last property above can be restated as,

6. 6 Distribution Function pmf: defined for a specific rv value, i.e., Probability of a set  Cumulative Distribution Function (CDF)

7. 7 Distribution Function properties

8. 8 Equivalence:  Probability mass function  Discrete density function (consider integer valued random variable) cdf: pmf: Discrete Random Variables

9. 9 Common discrete random variables Constant Uniform Bernoulli Binomial Geometric Poisson

10. 10 Constant Random Variable pmf CDF c 1.0 c

11. 11 Discrete Uniform Distribution Discrete rv X that assumes n discrete value with equal probability 1/n Discrete uniform pmf Discrete uniform distribution function:

12. 12 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 :

13. 13 Bernoulli Distribution CDF x 0.01.0 q p+q=1

14. 14 Binomial Random Variable Binomial rv  a fixed no. n of Bernoulli trials (BTs) RV Y n : no. of successes in n BTs Binomial pmf b(k;n,p) Binomial CDF

15. 15 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 X i ’s are independent identically distributed Bernoulli random variables.

16. 16 Binomial Random Variable: pmf pkpk

17. 17 Binomial Random Variable: CDF

18. 18 Reliability of a k out of n system Series system: Parallel system: Applications of the binomial

19. 19 Transmitting an LLC frame using MAC blocks p is the prob. of correctly transmitting one block Let p K (k) be the pmf of the rv K that is the number of LLC transmissions required to transmit n MAC blocks correctly; then Applications of the binomial

20. 20 Geometric Distribution Number of trials upto and including the 1 st success. In general, S may have countably infinite size Z has image {1,2,3,….}. Because of independence,

21. 21 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

22. 22 Geometric Distribution (contd.) Z rv: total no. of trials upto and including the 1 st success. Modified geometric pmf: does not include the successful trial, i.e. Z=X+1. Then X is a modified geometric random variable.

23. 23 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 …. Applications of the geometric

24. 24 Negative Binomial Distribution RV T r : no. of trials until r th success. Image of T r = {r, r+1, r+2, …}. Define events:  A: T r = n  B: Exactly r-1 successes in n-1 trials.  C: The n th trial is a success. Clearly, since B and C are mutually independent,

25. 25 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

26. 26 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.

27. 27 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 > 0 is the expected number of event occurrences per unit time 2.The number of events in two non-overlapping intervals are mutually independent

28. 28 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.

29. 29 Poisson Failure Model (cont.) The successive interevent times X 1, X 2, … 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 X 1, which has an exponential distribution The mean interevent time is 1/, which in this case is the mean time to failure

30. 30 Probability mass function (pmf) (or discrete density function): Distribution function (CDF): Poisson Distribution

31. 31 p k t=1.0 Poisson pmf

32. 32 t123456789 10 0.5 0.1 CDF 1 t=1.0 Poisson CDF

33. 33 t=4.0 p k t=4.0 Poisson pmf

34. 34 t CDF 123456789 10 0.5 0.1 1 t=4.0 Poisson CDF

35. 35 Probability Generating Function (PGF) Helps in dealing with operations (e.g. sum) on rv’s Letting, P(X=k)=p k, PGF of X is defined by, One-to-one mapping: pmf (or CDF)  PGF See page 98 for PGF of some common pmfs

36. 36 Discrete Random Vectors Examples:  Z=X+Y, ( X and Y are random execution times )  Z = min(X, Y) or Z = max(X 1, X 2,…,X k ) X:(X 1, X 2,…,X k ) is a k-dimensional rv defined on S  For each sample point s in S,

37. 37 Discrete Random Vectors (properties)

38. 38 Independent Discrete RVs X and Y are independent iff the joint pmf satisfies: Mutual independence also implies: Pair wise independence vs. set-wide independence

39. 39 Discrete Convolution Let Z=X+Y. Then, if X and Y are independent, In general, then,