Random Processes in Communication and Control/M-Sep14

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Last Time[edit]




c) event

Some Useful Random Variables[edit]

Bernoulli R.V[edit]

success probability


1) Flip a coin # of H

2) Manufacture a Chip # of acceptable chips

3) Bits you transmit successfully by a modem

Geometric Random Variable[edit]

Number of trials until (and including) a success for an underlying Bernoulli


1) Repeated coin flips # of tosses until H

2) Manufacture chips 3 of chips produced until an acceptable time

Binomial R.V[edit]

"# of successes in n trials"


1) Flip a coin n times. # of heads.

2) Manufacture n chips. # of acceptable chips.

Note: Binomial where are independent Bernoulli trials

Note: n=1; Binomial=Bernoulli;

Pascal R.V[edit]

"number of trials until (and including) the kth success with an underlying Bernoulli"

where is successes in trials

Note: Pascal where are geometric R.V.

Note: K=1 Pascal=Geometric


# of flips until the kth H

Discrete Uniform R.V.[edit]


1) Rolling a die.

2) Flip a fair coin. =# of H

Poisson R.V.[edit]

(Exercise) limiting case of binomial with

PMF is a complete model for a random variable

Cumulative Distribution Function[edit]

Like PMF, CDF is a complete description of random variable.


Flip the coins # of H

Properties of CDF[edit]

  • a)

"starts at 0 and ends at 1"

  • b) For all ,

"non-decreasing in x"

  • c) For all

"probabilities can be found by difference of the CDF"

  • d) For all ,

"CDF is right continuous"

  • e) For

"For a discrete random variable, there is a jump (discontinuity) in the CDF at each value . This jump equals

  • f) for all

"Between two jumps the CDF is constant"

  • g)

Continuous Random Variables[edit]

outcomes uncountable many


T: arrival of a partical

V: voltage

: angle

: distance



For any random variable (continuous or discrete)

  • a)

  • b) is nondecreasing in

  • c)

  • d) is right continuous


where A, B are intervals of the same length contained in [0,1]


Probability Density Function (PDF)[edit]

discrete: PMF <--> CDF (sum/difference)

continuous <---> (derivative/integral)

Theorem: Properties of PDF[edit]

  • a) ( is nondecreasing)

  • b)

  • c)


Some useful continuous Random Variables[edit]

Uniform R.V[edit]

Exponential R.V[edit]

Gaussian (Normal) R.V.[edit]