I would like to transform a standard normal random variable. How can I find and apply a transformation to make X an exponentially distributed random variable Y, and also into a Poisson distributed random variable Z?
Thanks!
Standard Normal Random Variable Transform
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0How about one hwk question at a time? Do you want to sample standard normal via random uniform observations? Then see the Box-Muller method in Wilipedia (or your text). To get exponential or Poisson from uniform you need the inverse CDF method. Both methods are likely in your text, or (with widely varying reliability) on the Internet. – 2017-02-05
1 Answers
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If you have the cumulative distribution function of a standard normal $\Phi(x)$ (e.g. pnorm in R or NORM.S.DIST in MSExcel) then it is relatively easy:
$$X' = - \dfrac1\lambda \log\left(1-\Phi(X)\right)$$
and leaving out the "$1-$" would also work due to symmetry. In effect this transforms in two steps via a uniform random variable on $[0,1]$
In R, using Xprime <- qexp(pnorm(X), rate=lambda)
would be even simpler and similar transformations would work for other continuous distributions