Random Processes in Communication and Control/M-Sep14

Last Time PMF a) b) c) event Some Useful Random Variables Bernoulli R.V success probability Example 1) Flip a coin # of H 2) Manufacture a Chip # of acceptable chips 3) Bits you transmit successfully by a modem Geometric Random Variable Number of trials until (and including) a success for an underlying Bernoulli Example 1) Repeated coin flips # of tosses until H 2) Manufacture chips 3 of chips produced until an acceptable time Binomial R.V "# of successes in n trials" Example 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 "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 # of flips until the kth H Discrete Uniform R.V. Example 1) Rolling a die. 2) Flip a fair coin. =# of H Poisson R.V. (Exercise) limiting case of binomial with PMF is a complete model for a random variable Cumulative Distribution Function Like PMF, CDF is a complete description of random variable. Example Flip the coins # of H Properties of CDF 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 outcomes uncountable many Example T: arrival of a partical V: voltage : angle : distance No PMF, Theorem For any random variable (continuous or discrete) a) b) is nondecreasing in c) d) is right continuous Example where A, B are intervals of the same length contained in [0,1] (exercise) Probability Density Function (PDF) discrete: PMF <--> CDF (sum/difference) continuous <---> (derivative/integral) Theorem: Properties of PDF a) ( is nondecreasing) b) c) Theorem Some useful continuous Random Variables Uniform R.V Exponential R.V Gaussian (Normal) R.V.