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I have been given an algorithm to generate a random variable $Y$, as follows:

  • $X=-\ln(Z)$
  • If $X>2$ then $Y=X-2$, otherwise reject.

I'm then asked to find the pdf and cdf of $Y$.

Can anyone tell me how to read this?

I know in theory what both are but I'm finding it difficult to infer them from problems like this.

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    $\begingroup$ I think you will need to tell us what $Z$ is first. $\endgroup$ – whuber Feb 3 '14 at 19:36
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    $\begingroup$ Is $Z$ intended to be uniform on $(0,1)$? Or something else? $\endgroup$ – Glen_b -Reinstate Monica Feb 3 '14 at 22:31
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If $Z$ is intended to be standard uniform, you can use elementary probability arguments in terms of the cdf to derive that $-\ln Z$ is standard exponential (the details of which arguments I leave to you).

The distribution is then truncated to be $>2$ and then shifted by subtracting 2. For the truncation you just need to consider the effect on the density:

enter image description here

Or you can get there from the memoryless property of the exponential.

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