Random Processes in Communication and Control/M-Sep14

Last TimeEdit

PMF  


a)  


b)  


c) event  


 

Some Useful Random VariablesEdit

Bernoulli R.VEdit

 


  success probability

ExampleEdit

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


 

ExampleEdit

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"


 


ExampleEdit

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

ExampleEdit

 # of flips until the kth H

Discrete Uniform R.V.Edit

 


ExampleEdit

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.

ExampleEdit

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


ExampleEdit

T: arrival of a partical


 


V: voltage


 


 : angle


 


 : distance


 


 


No PMF,  

TheoremEdit

For any random variable (continuous or discrete)


  • a)  


  • b)   is nondecreasing in  


  • c)  


  • d)   is right continuous


ExampleEdit

 


  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 PDFEdit

  • a)   (  is nondecreasing)


  • b)  


  • c)  

TheoremEdit

 

Some useful continuous Random VariablesEdit

Uniform R.VEdit

 


Exponential R.VEdit

 


 


Gaussian (Normal) R.V.Edit