EE255/CPS226 Discrete Random Variables

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EE255/CPS226Discrete Random Variables

• Dept. of Electrical Computer engineering
• Duke University
• Email bbm_at_ee.duke.edu, kst_at_ee.duke.edu

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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

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Probability Mass Function (pmf)

• Ax set of all samples (events) such that,
• pmf

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Pmf Properties

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

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Distribution Function

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

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Distribution Function (contd.)

• For integer valued X,

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Some common discrete distributions

• Constant
• Discrete Uniform
• Bernoulli
• Binomial
• Geometric
• Poisson

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Constant Random Variable

1.0
c
1.0
c
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Discrete Uniform Distribution

• Discrete rv X that assumes n discrete value with
  equal probability 1/n
• Discrete uniform pmf
• Discrete uniform distribution function

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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
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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

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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,

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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

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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.

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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,

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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

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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.

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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

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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,

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Discrete Random Vectors (properties)

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Independent Discrete RVs

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

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Discrete Convolution

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