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}. $
