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Unformatted text preview: Probability and Statistics with Reliability, Queuing and Computer Science Applications: second edition by K.S. Trivedi Publisher-John Wiley & Sons Chapter 2:Discrete Random Variables Dept. of Electrical & Computer engineering Duke University Email: kst@ee.duke.edu URL: www.ee.duke.edu/~kst Copyright © 2006 by K.S. Trivedi 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 of ‘2n’ to size of just ‘n+1’. Such abstractions lead to the notion of a random variable. Copyright © 2006 by K.S. Trivedi 2 Random Variables (cont’d) Discrete RV X: countable number of values. – Properties of a discrete RV: – Probability mass function: p(xi) of X Continuous RV X: uncountably infinite number of different values (an interval or a collection of intervals). (In Chapter 3) – Properties of a continuous RV: – Probability density function : f(x) of X Note the distinction: pmf vs. pdf Notation: uppercase letters (X, Y) for RVs, lowercase, x,y, for the values. Copyright © 2006 by K.S. Trivedi 3 Discrete Random Variables A random variable (rv) X is a mapping (function) from the sample space S to the set of real numbers. i.e., a function that assigns a real number to each sample point Random variable is not a variable but a function If image of X (set of all values taken by X), is finite or countably infinite, X is a discrete rv. Copyright © 2006 by K.S. Trivedi 4 Discrete Random Variables Inverse image Ax 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 the set of all inverse images are set that is mutually exclusive and collectively exhaustive: Copyright © 2006 by K.S. Trivedi 5 Discrete Random Variables (Example 2.1) Consider a random experiment defined by a sequence of three Bernoulli trials. The sample space S consists of eight triples of 0s and 1s Define a random variable X to be the total number of successes from three trials The values of the random variable X are {0,1,2,3} X(0,0,0) =0 X(0,0,1)=X(0,1,0)=X(1,0,0)=1 X(0,1,1)=X(1,0,1)=X(1,1,0) =2 X(1,1,1) = 3 The inverse images of random variable X are A0 = {(0,0,0)} ; A1={(0,0,1),(0,1,0),(1,0,0)}; A2={(0,1,1),(1,1,0),(1,0,1)}; A3 = {(1,1,1)} Copyright © 2006 by K.S. Trivedi 6 Probability Mass Function (pmf) Ax : set of all sample points such that, pmf is defined as = the probability that the value of the random variable X obtained on a performance of the experiment is equal to x Copyright © 2006 by K.S. Trivedi 7 Equivalence pmf: Probability mass function Discrete density function Sometimes mistakenly called the distribution function Empirical version is called a histogram Copyright © 2006 by K.S. Trivedi 8 pmf Properties The following properties hold for pmf , since the random variable assigns some value to each sample point Since a discrete rv X takes a finite or a countably infinite set values, , the last property above can be restated as, Copyright © 2006 by K.S. Trivedi 9 Cumulative Distribution Function Note that pmf is defined for a specific rv value, i.e., Probability of a set The function defined as is called the cumulative distribution function (CDF) or the probability distribution function or simply the distribution function of the random variable X Copyright © 2006 by K.S. Trivedi 10 Equivalence & Notes CDF (cumulative distribution function) PDF (probability distribution function) not recommended as confusion with pdf (prob. density function) may arise Distribution function FX(t) , subscript indicates the name of the rv; it must be a capital letter; the subscript may be omitted if the rv is clear from the context; the argument is a dummy variable so FX(t) & FX(y) is the same function while FX(t) & FY(t) are different functions Copyright © 2006 by K.S. Trivedi 11 Distribution Function properties F(x) is a monotone increasing function of x, since if Copyright © 2006 by K.S. Trivedi 12 Common discrete random variables Constant Uniform Bernoulli Binomial Geometric Negative binomial Poisson Hypergeometric Copyright © 2006 by K.S. Trivedi 13 Constant Random Variable pmf 1.0 c CDF 1.0 c This a deterministic quantity (value); it may seem strange to call it an rv but often we need to mix deterministic and random quantities Copyright © 2006 by K.S. Trivedi 14 Constant Random Variable in SHARPE In order to assign this distribution to a block in rbd, to a basic event in fault tree or to a task in a task graph, we can only take two possible values of c: c=0 and c=∞. For other cases, as we will see later, approximations can be used Block try comp z zero * prob(0); time to failure is 0 comp I inf * prob(1); time to failure is ∞ … Copyright © 2006 by K.S. Trivedi 15 Discrete Uniform Distribution Discrete rv X that assumes n discrete values with equal probability 1/n Discrete uniform pmf Discrete uniform distribution function: assume X takes integer values 1,2,3,…,n then, for Copyright © 2006 by K.S. Trivedi 0≤ x≤n 16 Discrete Uniform pmf Copyright © 2006 by K.S. Trivedi 17 Discrete Uniform CDF Copyright © 2006 by K.S. Trivedi 18 Notation: Floor & Ceiling Define Copyright © 2006 by K.S. Trivedi 19 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: pX(1) = p1 = P(X=1) = p pX(0) = p0 = P(X = 0) = q = 1-p Copyright © 2006 by K.S. Trivedi 20 Bernoulli Distribution CDF p+q=1 q 0.0 1.0 x CDF of Bernoulli random variable Copyright © 2006 by K.S. Trivedi 21 Binomial Random Variable Binomial rv a fixed no. n of independent Bernoulli trials (BTs) RV Yn: no. of successes in n BTs. Binomial pmf b(k;n,p) Binomial CDF Copyright © 2006 by K.S. Trivedi 22 Regarding Parameters Note that the binomial pmf (or distribution) is completely defined by the formula given earlier, and by its "parameters" n and p.The binomial probability equation never changes so we regard a binomial distribution as being defined by its parameters. This is typical of all probability distributions (using their own parameters, of course). One of the problems we often face in statistics is estimating the parameters after collecting data that we know (or believe) comes from a particular probability distribution (such as the binomial). We will deal with parameter estimation in Chapter 10. Copyright © 2006 by K.S. Trivedi 23 Binomial Random Variable: pmf pk Copyright © 2006 by K.S. Trivedi 24 1.2 Binomial Random Variable: CDF 1 CDF 0.8 0.6 0.4 0.2 0 0 1 2 3 4 5 6 7 8 9 x Copyright © 2006 by K.S. Trivedi 25 10 Conditions for application of the binomial distribution Applicable wherever a series of trials is made such that Each trial has two mutually exclusive outcomes “success” and “failure” (or 0 and 1) The probability of success at each trial is a constant (therefore, probability of failure too) The outcomes of successive trials are mutually independent Copyright © 2006 by K.S. Trivedi 26 Applications of the binomial pmf Used frequently in quality control, reliability, survey sampling, and other industrial problems Typical situation where these conditions apply is where number of defective components are counted when several are selected from a large batch of components This situation can be generalized to many other applications as in the following slides Copyright © 2006 by K.S. Trivedi 27 Applications of the binomial pmf Reliability of a k out of n system, R is the reliability of an individual component 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) = ∑ (nj )[R] j [1 − R]n− j = [ R]n j =n Parallel system: n Rparallel = 1 − b(0; n, R) = ∑ (nj )[R] j [1 − R]n− j = 1 − [1 − R]n j =1 Copyright © 2006 by K.S. Trivedi 28 Binomial Random Variable in SHARPE Binomial CDF block bin(q, k, n) comp one prob(q) kofn block0 k, n, one end bind n 5 bind R 0.9 loop k,1,n expr sysprob(bin; 1-R, k, n) * B(k-1;n,R) is computed and printed end end Copyright © 2006 by K.S. Trivedi 29 Applications of the binomial dist. Transmitting an LLC frame using n 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 (1) = b(n; n, p ) = p n pK 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 Copyright © 2006 by K.S. Trivedi 30 Applications of the binomial dist. Counting the number of defective chips in a sample of size 35. 10% of them are expected to be defective (p = 0.1 is the prob. of success) The observed fraction defective should be close to the binomial pmf: The observed data and the binomial pmf are shown in the following table Copyright © 2006 by K.S. Trivedi 31 Observed Data Number of Defects Number of samples showing this number of defects Fraction (of 800 samples) showing this No. of defects 0 11 0.01375 1 95 0.11875 2 139 0.17375 3 213 0.26625 4 143 0.17875 5 113 0.14125 6 49 0.06125 7 27 0.03375 8 6 0.00750 9 4 0.00500 10 0 0.00000 Copyright © 2006 by K.S. Trivedi 32 Binomial pmf k = Number of Defects/sample Data b(k;35,0.1) 0 0.01375 1 0.11875 0.0974 2 0.17375 0.1839 3 0.26625 0.2248 4 0.17875 0.1998 5 0.14125 0.1376 6 0.06125 0.0765 7 0.03375 0.0352 8 0.00750 0.0137 9 0.00500 0.0046 10 0.00000 0.0013 Copyright © 2006 by K.S. Trivedi 0.0250 33 Comparing the model pmf with real data Com paring m odel pm f with data 0. 3 Fraction 0. 25 0. 2 Data 0. 15 b(k ; 35, 0. 1) 0. 1 0. 05 0 0 1 2 3 4 5 6 7 8 9 10 k Copyright © 2006 by K.S. Trivedi 34 Applications of the binomial dist. Transmitting binary digits through a communication channel, the number of digits received correctly, Cn, out of n transmitted digits ~ binomial distribution B(k;n;p), where p = prob. of successfully transmitting one digit. The prob. of exactly i errors Pe(i) and the prob. of an error-free transmission is given by: pe (i ) = pCn (n − i ) = ( in ) p ( n −i ) (1 − p ) i , and pe (0) = p n . Copyright © 2006 by K.S. Trivedi 35 Applications of the binomial dist. Taking a random sample of 10 VLSI chips from a very large batch. The No. of defective chips in the sample has the pmf b(k;10,p), where p = Prob. of a randomly chosen chip is defective. No defective in the sample -> accept the batch Find defective in the sample -> reject the batch P (“a batch is accepted”) = P (“No defectives”) =(1-p)10. Copyright © 2006 by K.S. Trivedi 36 Computation of the binomial pmf Computation using the formula directly is numerically unstable A recursive formula should be used Normal or Poisson approximation can be used; see page 75 of the blue book for details Pmf can be symmetrical (for p=0.5;fig. I in slide #33), positively skewed (for p < 0.5; fig. II in slide # 33) or negatively skewed (for p > 0.5; fig. III in slide # 34) Copyright © 2006 by K.S. Trivedi 37 Fig I: Symmetrical binomial pmf Fig II: Positively Skewed Binomial pmf Copyright © 2006 by K.S. Trivedi 38 Fig III: Negatively Skewed binomial pmf Copyright © 2006 by K.S. Trivedi 39 Binomial Random Variable We shall see later that 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. Copyright © 2006 by K.S. Trivedi 40 Geometric Distribution Consider a sequence of Bernoulli trials up to and including the 1st success. Sample space S is countably infinite in size If p is the probability of a success and q be the probability of failure of each Bernoulli trial (recall independence) . Then, rv Z has a geometric distribution with resp. pmf & CDF Copyright © 2006 by K.S. Trivedi 41 Geometric pmf Example Copyright © 2006 by K.S. Trivedi 42 Geometric CDF Example Copyright © 2006 by K.S. Trivedi 43 Modified Geometric Distribution Now let X be a rv counting total no. of trials upto but not including the 1st success. Modified geometric random variable: Defined as total no. of failed trials before first success, i.e., Z=X+1. Then X is a modified geometric random variable with pmf : Copyright © 2006 by K.S. Trivedi 44 Note Many well known papers don’t clearly distinguish between geometric and modified geometric distributions Be careful so that the right formula is used Copyright © 2006 by K.S. Trivedi 45 Geometric Distribution (contd.) Geometric distribution (and its cousins) is the only discrete distribution that exhibits MEMORYLESS property. Future outcomes are independent of the past events. Let Z be the rv denoting total number of trials upto and including the first success Assume n trials completed with all failures. Let Y denote additional trials upto and including the first success, i.e., Z = n+Y or Y=Z-n The conditional probability qi is given by Copyright © 2006 by K.S. Trivedi 46 Geometric Distribution (contd.) Thus after n unsuccessful trials, the number of trials remaining until the first success has the same pmf as Z had originally (i.e., memoryless property) Copyright © 2006 by K.S. Trivedi 47 Applications of the Geometric Dist. Consider the scheduling of a computer system with a fixed time slice. At the end of a time slice a program would either have completed execution with probability p or would need more computation with probability q = 1 – p . The random variable denoting the number of time slices needed to complete the execution of a program is geometrically distributed. The number of times the following statement is executed: repeat S until B is geometrically distributed assuming that successive tests of condition B satisfy the conditions of Bernoulli trials Copyright © 2006 by K.S. Trivedi 48 Applications of the Geometric Dist. The number of times the following statement is executed: while (¬B) do S is modified geometrically distributed assuming that successive tests of condition B satisfy the conditions of Bernoulli trials. Copyright © 2006 by K.S. Trivedi 49 Negative Binomial Distribution RV Tr: no. of trials up to and including the 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, Where, n = r, r+1, r+2, …… This pmf is known as negative binomial pmf Copyright © 2006 by K.S. Trivedi 50 Poisson Random Variable RV such as “no. of arrivals in an interval (0,t]” Assuming λ is rate of arrival of the job In a small interval, ∆t, prob. of new arrival= λ∆t. If ∆t is small enough, then probability of two arrivals in ∆t may be neglected Suppose the interval (0,t] is divided into n subintervals of length t/n Suppose arrival of a job in any interval is independent of the arrival of a job in any other interval . Copyright © 2006 by K.S. Trivedi 51 Poisson Random Variable For a large n, the n intervals can be thought of as constituting a sequence of Bernoulli trials with probability of success p = λt/n Therefore, probability of k arrivals in a total of n intervals, is given by As , this gives the Poisson pmf : Copyright © 2006 by K.S. Trivedi 52 Poisson Random Variable (contd.) Poisson rv often occurs in situations, such as, “no. of packets (or calls) arriving in t sec.” or “no. of components failing in t hours” etc. Copyright © 2006 by K.S. Trivedi 53 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: (λt ) P{N (t ) = k } = ⎡ ⎢ ⎣ k − λt / k!⎤ e ⎥ ⎦ k = 0, 1, 2, … Where λ > 0 is the expected number of events (failures) occurrences per unit time. 2. The number of events in two non-overlapping intervals are mutually independent. Copyright © 2006 by K.S. Trivedi 54 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. We will study stochastic processes in Chapter 6 and beyond. Copyright © 2006 by K.S. Trivedi 55 Poisson Failure Model (contd.) The successive interevent times X1, X2, … in a homogenous Poisson model, are mutually independent, and have a common exponential distribution given by: P {X 1 ≤ t } = 1 − e − λ t t≥0 To show this: P ( X 1 > t ) = P ( N ( t ) = 0) = e − λt Thus, the discrete random variable, N(t), with the Poisson distribution, is related to the continuous random variable X1, which has an exponential distribution (See Chapter 3). The mean interevent time is 1/λ, which in this case is the mean time to failure (See Chapter 4). Copyright © 2006 by K.S. Trivedi 56 Poisson Random Variable Probability mass function (pmf) (or discrete density function): (λt)k pk = P{N(t) = k} = e−λt k! Distribution function (CDF): k ⎣ x ⎦ −λt (λt ) F ( x) = ∑ e k =0 k! Copyright © 2006 by K.S. Trivedi 57 Poisson pmf pk λt=1.0 Copyright © 2006 by K.S. Trivedi 58 Poisson CDF CDF 1 λt=1.0 0.5 0.1 1 2 3 4 5 6 7 Copyright © 2006 by K.S. Trivedi 8 9 10 59 t pk Poisson pmf λt=4.0 λt=4.0 Copyright © 2006 by K.S. Trivedi 60 Poisson CDF CDF 1 λt=4.0 0.5 0.1 1 2 3 4 5 6 7 Copyright © 2006 by K.S. Trivedi 8 9 10 61 t Poisson Approximation Example The prob. of defective VLSI chip = 0.01. Find Prob(no defective chip in a box of 100 chips) Binomial pmf (n=100, p=0.01) b(k ; n, p ) = b(0;100,0.01) = (100 )(0.01) 0 (0.99)100 = 0.366. 0 Poisson approximation (α=np=100x0.01=1) f (k ; α ) = f (0;1) = e −1 = 0.3679. Copyright © 2006 by K.S. Trivedi 62 Another Poisson Example Connections arrive at a switch at a rate of 12 per ms. The arrival distribution is Poisson. (a) What is the probability that exactly 12 calls arrive in one ms? (b) What is the probability that exactly 100 calls arrive in 10 ms? (c) What is the probability that the number of calls arriving in 2 ms is greater that 7 and less than or equal to 10? Work it out yourself Copyright © 2006 by K.S. Trivedi 63 Computing Poisson pmf Direct computation of Poisson probabilities is numerically unstable A recommended recursive formula is: f (k + 1;α ) = αf (k ;α ) /(k + 1) With the intialization step: f (0;α ) = exp(−α ) But even this formula is unstable for very large values of α. See Fox 1988 for an algorithm Copyright © 2006 by K.S. Trivedi 64 Hypergeometric pmf Obtained when sampling without replacement (binomial pmf resulted when sampling was done with replacement) Probability of choosing k defective components in a random sample of m components, chosen without replacement, from a total of n components, d of which are defective, is given by the hypergeometric pmf , h(k;m,d,n) as Copyright © 2006 by K.S. Trivedi 65 Hyper Geometric PMF: Example 2.11 Compute the probability of obtaining 3 defectives in a sample of size 10 taken without replacement from a box of 20 components containing 4 defectives. Applying the formula with k=3, m=10, d=4, n=20, we get: If we approximated this probability using a binomial distribution with corresponding parameters, we would have obtained b(3;10,0.20)=0.2013, which is a considerable underestimate of the actual value. Copyright © 2006 by K.S. Trivedi 66 Hyper Geometric PMF: Example 2.12 Cellular Wireless System with TDMA: In this system base transceiver of each cell has n base repeaters, each of which provide m time-division-multiplexed channels. Let k be the total number of talking channels in the system, which are allocated randomly to the users. Now, if a base repeater fails, then the probability that i talking channels reside in the failed repeater is given by pi =h(i;k,m,mn). Copyright © 2006 by K.S. Trivedi 67 Hypergeomteric pmf (Example 2.13) Consider a s/w reliability growth model for estimating the number of residual faults in the s/w after testing phase A s/w is subjected to a sequence of n test instances ti, i=1,2,..n. Faults detected at each test are removed without inserting new ones. Number of faults detected by ti is denoted by Ni The cumulative number of faults detected by test instances from t1 thru ti is given by the random variable i.e. number of faults still undetected after ith test instance is m- Ci Copyright © 2006 by K.S. Trivedi 68 Hypergeomteric pmf (Example 2.13) The probability that k faults are detected by the test instance ti+1 given that ci faults are detected by test instances t1 through ti is Copyright © 2006 by K.S. Trivedi 69 Probability Generating Function (PGF) Helps in dealing with operations (e.g., sum) on nonnegative integer-valued 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 We will return to this topic in Chapter 4 Copyright © 2006 by K.S. Trivedi 70 PGFs of Some Well-known Distributions Bernoulli R.V. : Binomial R.V. : Modified Geometric R.V. : Poisson R.V. : Uniform R.V. : Constant R.V. : Copyright © 2006 by K.S. Trivedi 71 PGF and Distributions Thm 2.1: If two discrete R.V.s X and Y have same PGFs, then they must have the same distributions and pmfs. Copyright © 2006 by K.S. Trivedi 72 Application of PGF: Recurrence Relation for Binomial pmf Differentiating both sides we get Equating the coefficients of zk-1 on each side we get, Copyright © 2006 by K.S. Trivedi 73 Application of PGF: Another Recurrence Relation for Binomial pmf Multiplying each side with (q+pz), we get: Equating coefficients of zk on each sides we get: Copyright © 2006 by K.S. Trivedi 74 Application of PGF: Recurrence Relation for Poisson pmf Differentiating both sides and equating the coefficients of zk on both the sides we get: Copyright © 2006 by K.S. Trivedi 75 Discrete Random Vectors X:(X1, X2,…,Xr) be r rv defined on a sample space S For each sample point s in S, each of the rv X1, X2,…,Xr takes on one of its possible values, as The random vector X = (X1, X2,…,Xr) is an r-dimensional vector – valued function Copyright © 2006 by K.S. Trivedi 76 Joint or compound pmf (properties) The joint or compound pmf of a random vector X is defined to be The properties of this pmf are : Copyright © 2006 by K.S. Trivedi 77 Joint or compound pmf (Example) An interesting example of a compound pmf is the multinomial pmf Consider a sequence of generalized Bernoulli trials, with r distinct outcomes in each trial with probabilities , where Define a random vector X = (X1, X2,…,Xr), such that Xi is the number of trials that resulted in the ith outcome. Then the compound pmf of X is given by Copyright © 2006 by K.S. Trivedi 78 Joint or compound pmf (Example) The marginal pmf of Xi may be computed by summing the joint pmf over all nj’s except ni. The marginal pmf of each Xi is binomial with parameters n and pi. Consider the case when a program requires I/O service from device i with probability pi at the end of a CPU burst, with If n CPU bursts are observed, then the probability that ni of these will be directed to I/O device i (for i = 1,2,…,r) is given by the multinomial pmf. The number of I/O requests (out of n) directed to a specific device j, has a binomial distribution with parameters n and pj Copyright © 2006 by K.S. Trivedi 79 Independent Discrete Random Variables X and Y are independent iff the joint pmf satisfies: Mutual independence also implies: Pair wise independence vs. mutual independence It is possible for every pair of random variables in the set {X1,X2,…,Xr} to be pairwise independent without the entire set being mutually independent Copyright © 2006 by K.S. Trivedi 80 Discrete Convolution Let Z=X+Y . Then, if X and Y are independent, t pZ (t ) = p X +Y (t ) = ∑ p X ( x ) pY (t − x ) x =0 This sum is known as the discrete convolution Copyright © 2006 by K.S. Trivedi 81 Discrete Convolution Given events X and Y are independent, consider the probability of event Z=X+Y=t On a two dimensional(x,y) event space, this event is represented by all the event points on the line X+Y=t (as shown in the figure). The probability of Z can be computed by adding event points on this line. Hence, Copyright © 2006 by K.S. Trivedi 82 Discrete Convolution (Contd.) This summation is called Discrete Convo -lution and it gives the formula for the pmf of the sum of two nonnegative independent discrete random variables Copyright © 2006 by K.S. Trivedi 83 Discrete Convolution Let Z=X+Y . Then, if X and Y are independent, t pZ (t ) = p X +Y (t ) = ∑ p X ( x ) pY (t − x ) x =0 This sum is known as the discrete convolution Restricting to nonnegative integer-valued random variables, using the probability generating function (PGF), the PGF of Z can be represented as In general, then, if Xi‘ s are independent Copyright © 2006 by K.S. Trivedi 84 Some results on Sum of discrete independent random variables Let X1,X2,…,Xr be mutually independent If Xi is binomially distributed with parameters ni and p, then sum of Xis i.e. has the binomial distribution with parameters and p If Xi has the (modified) negative binomial distribution with parameters αi and p, then has the (modified) negative binomial distribution with parameters and p If Xi has the Poisson distribution with parameter αi, then has the Poisson distribution with parameter Copyright © 2006 by K.S. Trivedi 85 Theorem 2.2: Closure of Distributions Under Sum Copyright © 2006 by K.S. Trivedi 86 ...
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This note was uploaded on 04/08/2010 for the course COMPUTER E 409232 taught by Professor Mohammadabdolahiazgomiph.d during the Spring '10 term at Islamic University.

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