Random Processes in Communication and Control/M-Sep14

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Last Time[edit]

PMF


a)


b)


c) event


Some Useful Random Variables[edit]

Bernoulli R.V[edit]


success probability

Example[edit]

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


Example[edit]

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"



Example[edit]

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

Example[edit]

# of flips until the kth H

Discrete Uniform R.V.[edit]


Example[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.

Example[edit]

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


Example[edit]

T: arrival of a partical



V: voltage



: angle



: distance




No PMF,

Theorem[edit]

For any random variable (continuous or discrete)


  • a)


  • b) is nondecreasing in


  • c)


  • d) is right continuous


Example[edit]


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





(exercise)


Probability Density Function (PDF)[edit]


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


continuous <---> (derivative/integral)


Theorem: Properties of PDF[edit]

  • a) ( is nondecreasing)


  • b)


  • c)

Theorem[edit]

Some useful continuous Random Variables[edit]

Uniform R.V[edit]


Exponential R.V[edit]



Gaussian (Normal) R.V.[edit]