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.

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

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

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

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

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

## Alumni Liaison

Basic linear algebra uncovers and clarifies very important geometry and algebra.

Dr. Paul Garrett