Random Processes in Communication and Control/M-Sep14

Last TimeEdit




c) event  


Some Useful Random VariablesEdit

Bernoulli R.VEdit


  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 VariableEdit

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.VEdit

"# 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.VEdit

"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 FunctionEdit


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


Flip the coins  # of H



Properties of CDFEdit

  • 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 VariablesEdit

outcomes uncountable many


T: arrival of a partical


V: voltage


 : angle


 : distance



No PMF,  


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 PDFEdit

  • a)   (  is nondecreasing)

  • b)  

  • c)  



Some useful continuous Random VariablesEdit

Uniform R.VEdit


Exponential R.VEdit



Gaussian (Normal) R.V.Edit