random - CDA6530: Performance Models of Computers and...

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CDA6530: Performance Models of Computers and Networks Chapter 2: Review of Practical Random Variables
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2 Definition Random variable (R.V.) X: A function on sample space X: S R Cumulative distribution function (CDF): Probability distribution function (PDF) Distribution function F X (x) = P(X<x) Probability density function (pdf): Used for continuous R.V. F X ( x )= R x −∞ f X ( t ) dt f X ( x dF X ( x ) dx
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3 Two Classes of R.V. Discrete R.V. Bernoulli Binomial Geometric Poisson Continuous R.V. Uniform Exponential, Erlang Normal Closely related Exponential  Geometric Normal  Binomial, Poisson
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4 Bernoulli A trial/experiment, outcome is either “success” or “failure”. X=1 if success, X=0 if failure P(X=1)=p, P(X=0)=1-p Bernoulli Trials A series of independent repetition of Bernoulli trial.
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5 Binomial A Bernoulli trials with n repetitions Binomial: X = No. of successes in n trails X B(n, p) P ( X = k ) f ( k ; n, p )= à n k ! p k (1 p ) n k where à n k ! = n ! ( n k )! k !
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6 Binomial Example (1) A communication channel with (1-p) being the probability of successful transmission of a bit. Assume we design a code that can tolerate up to e bit errors with n bit word code. Q: Probability of successful word transmission? Model: sequence of bits trans. follows a Bernoulli Trails Assumption: each bit error or not is independent P(Q) = P(e or fewer errors in n trails) = P e i =0 f ( i ; n, p ) = P e i =0 Ã n i ! p i (1 p ) n i
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Introduction 1-7 Binomial Example (2) ---- Packet switching versus circuit switching 1 Mb/s link each user: 100 kb/s when “active” active 10% of time circuit-switching: 10 users packet switching: with 35 users, prob. of > 10 active less than .0004 Packet switching allows more users to use network! N users 1 Mbps link Q: how did we know 0.0004?
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8 Geometric Still about Bernoulli Trails, but from a different angle. X: No. of trials until the first success Y: No. of failures until the first success P(X=k) = (1-p) k-1 p P(Y=k)=(1-p) k p X Y
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9 Poisson Limiting case for Binomial when: n is large and p is small n>20 and p<0.05 would be good approximation Reference: wiki λ =np is fixed, success rate X: No. of successes in a time interval (n time units) Many natural systems have this distribution The number of phone calls at a call center per minute. The number of times a web server is accessed per minute. The number of mutations in a given stretch of DNA after a certain amount of radiation.
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random - CDA6530: Performance Models of Computers and...

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