20161ee232B_1_232B HW4 02 04 2016A - UCLA Electrical...

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1 UCLA Electrical Engineering Department EE232B HW Set #4 Professor Izhak Rubin Problem 1 Note: the moment generating function of the sum of independent random variables is equal to the product of their moment generating functions. Using the above knowledge to show the following: (a) The sum of two independent Poisson random variables is a Poisson random variable. (b) The sum of two independent and identical exponential random variables has a Gamma (or Erlang) distribution. (c) The sum of two independent but non-identical exponential random variables has a probability density function that can be represented as a (weighted) linear combination of the two exponential densities. Problem 2 For M/M/1 and M/M/m, find E[W | W > 0], that is, the expected time that one must wait in the queue, given that one has to wait at all. Problem 3 For M/M/1, when at steady state, derive the message delay CDF D(t) and pdf d(t), showing them to be given by the equations:     1 0 0 t t D t e t d t e t
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  • Winter '16
  • IzhakRubin
  • Probability theory, Professor Izhak Rubin, exponential random variables, UCLA Electrical Engineering Department

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