ECE Ph.D. Qualifying Exam

Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

August 2003



2. (15% of Total)

You want to simulate outcomes for an exponential random variable $ \mathbf{X} $ with mean $ 1/\lambda $ . You have a random number generator that produces outcomes for a random variable $ \mathbf{Y} $ that is uniformly distributed on the interval $ \left(0,1\right) $ . What transformation applied to $ \mathbf{Y} $ will yield the desired distribution for $ \mathbf{X} $ ? Prove your answer.

Answer

$ f_{\mathbf{X}}\left(x\right)=\lambda e^{-\lambda x}. $

$ F_{\mathbf{X}}\left(x\right)=1-e^{-\lambda x}. $

$ y=1-e^{-\lambda x} $
$ e^{-\lambda x}=1-y $
$ -\lambda x=\ln\left(1-y\right) $
$ x=\frac{-\ln\left(1-y\right)}{\lambda} $
$ x=\frac{-\ln y}{\lambda}. $

$ \mathbf{X}=F_{\mathbf{X}}^{-1}\left(\mathbf{Y}\right). $

$ F_{\mathbf{X}}^{-1}\left(y\right)=\frac{-\ln y}{\lambda}. $

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