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EE255/CPS226 Discrete Random Variables

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Geometric distribution is the only discrete distribution that exhibits MEMORY-LESS property. ... n trials completed with all failures. ... – PowerPoint PPT presentation

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Title: EE255/CPS226 Discrete Random Variables


1
EE255/CPS226Discrete Random Variables
  • Dept. of Electrical Computer engineering
  • Duke University
  • Email bbm_at_ee.duke.edu, kst_at_ee.duke.edu

2
Discrete Random Variables
  • A random event is not necessarily a number e.g
  • (T,F), (R,Y,B), (american, british,..,zimbabwe)
    etc.
  • A random variable (rv) X is a mapping (function)
  • X is more accurately known as the IMAGE of s
  • Inverse image

3
Probability Mass Function (pmf)
  • Ax set of all samples (events) such that,
  • pmf

4
Pmf Properties
  • If X is a finite or a countably infinite set
    values,
  • above property gets redefined as,

5
Distribution Function
  • pmf defined for a specific rv value, i.e.,
  • Probability of a set
  • Cumulative Distrib Func (CDF)

6
Distribution Function (contd.)
  • For integer valued X,

7
Some common discrete distributions
  • Constant
  • Discrete Uniform
  • Bernoulli
  • Binomial
  • Geometric
  • Poisson

8
Constant Random Variable

1.0
c
1.0
c
9
Discrete Uniform Distribution
  • Discrete rv X that assumes n discrete value with
    equal probability 1/n
  • Discrete uniform pmf
  • Discrete uniform distribution function

10
Bernoulli Distribution
  • Bernoulli distribution ? RV generated by a single
    Bernoulli trial that has a binary valued outcome
    0,1

1.0
Pq1
q
x
0.0
1.0
11
Binomial Distribution
  • Binomial distribution ? multiple Bernoulli trials
    (BTs)
  • RV Yn no. of successes in n BTs, i.e.
    1,0,0,1,0,1,1
  • Above is the equation for the Binomial pmf (p,n)
  • Binomial CDF

12
Geometric Distribution
  • Multiple Bernoulli trials ? occurrence of 1st
    success.
  • In general, S may have countably infinite size
  • Z has image 1,2,3,.. Assuming independent
    trials,

13
Geometric Distribution (contd.)
  • Geometric distribution is the only discrete
    distribution that exhibits MEMORY-LESS 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 nY or YZ-n

14
Geometric Distribution (contd.)
  • Z rv total no. of trials to 1st success. This
    count includes the successful trial.
  • Modified geometric pmf does not includes the
    successful trial, i.e. ZX1. Then X follows
    modified geometric distrbution.

15
Nagative Binomial Distributiond.)
  • RV Tr no. of trials until rth success.
  • Image of Tr r, r1, r2, . 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,

16
Poisson Distribution
  • RV such as no. of arrivals in an interval 0,t)
  • In a small interval, ?t, prob. of new arrival
    ??t.
  • pmf b(kn, ?t/n) CDF B(kn, ?t/n)
  • What happens when

17
Poisson Distribution (contd.)
  • Poisson distribution often occurs in situations,
    such as, no. of packets (or calls) arriving in
    t sec. or no. of components failing in t sec
    etc.

18
Probability Generating Function (PGF)
  • Helps in dealing with operations (e.g. , x) on
    rvs
  • Letting, P(Xk)pk , PGF of X is defined by,
  • One-to-one mapping pmf (or CDF) ??PGF
  • See page 98 for PGF of some common pmfs

19
Discrete Random Vectors
  • Examples
  • ZXY, (X and Y are random execution times)
  • Z min(X, Y) or Z max(X1, X2,,Xk)
  • X(X1, X2,,Xk) is a k-dim rv defined on S
  • For each sample point s in S,

20
Discrete Random Vectors (properties)

21
Independent Discrete RVs
  • X and Y are independent iff the joint pmf
    satisfies
  • Mutual independence also implies
  • Pair wise independence vs. set-wide independence

22
Discrete Convolution
  • Let ZXY . Then, if X and Y are independent,
  • In general, then,
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