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I'm having trouble deriving the PDF for the following problem (from the book Doing Bayesian analysis with R and BUGS):

Consider a spinner of the kind often found with board games, but with a log10 scale from 1 to 100 (covering 0 to 360 deg). The spinner is fair, so any value is equally likely. What is the pdf at point $x$?

I know that the $x$ value at angle $t$ is given by

$$x = 100^\text{t/360}$$

Doing some transformation

$$x = \exp^{(t/360) \times \log(100)}$$

$$pdf(x) = pm(x)/dx$$

I also know

$$dt/dx = 360/(\log100 \times x)$$

Question

How to derive the answer below, using the information above?

Book answer

$$pdf(x) = 1/(2 \times ln(10) \times x)$$

Thanks.

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  • $\begingroup$ Don't think in terms of degrees: it's a relatively arbitrary way of measuring locations on the spinner, anyway. If you do not find that your question is answered at stats.stackexchange.com/questions/14483/…, then please edit it to show us what additional help you need and we can re-open this thread. $\endgroup$ – whuber♦ Oct 7 '13 at 14:07
  • $\begingroup$ I've edited the question showing my ideas - i'll read the related question, thanks! $\endgroup$ – Fernando Oct 7 '13 at 14:47
  • $\begingroup$ Fernando, please add the self-study tag to the question. $\endgroup$ – COOLSerdash Oct 7 '13 at 15:02
  • $\begingroup$ Ok, new tag added. $\endgroup$ – Fernando Oct 7 '13 at 15:24
  • $\begingroup$ @whuber - could you please remove the duplicated tag? $\endgroup$ – Fernando Oct 7 '13 at 20:47