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I'm learning bayesian and I need to do some proofs that I'm not quite sure I can do. Any help will be very welcome!

If $\theta \text{~Unif}(0,1)$

What's the density of $y = - \log\theta$?

And what's density of $\theta \ (1-\theta)$?

Finally, what's density of $\log (\theta \ (1 - \theta))$?


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1 Answer 1

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All three are similar:

1) $-\log\theta$ is monotonically decreasing on $(0;1]$ from $+\infty$ to $0$, therefore, for $y\ge 0$: $F_{-\log\theta}(y)=Pr\{-\log\theta<y\}=Pr\{\theta>e^{-y}\}=1-e^{-y}$, and density is $e^{-y}$.

2) $\theta(1-\theta)$ is increasing on $[0;1/2]$ from $0$ to $1/4$, and decreasing on $[1/2;1]$ from $1/4$ to $0$, therefore, for $y\in[0;1/4]$: $F_{\theta(1-\theta)}(y)=Pr\{\theta(1-\theta)<y\}=1-Pr\{\theta(1-\theta)>y\}=1-Pr\{\theta^2-\theta+y<0\}=1-\sqrt{1-4y}$, and density is just the derivative.

3) Similarly. I am pretty sure you can now get it done by yourself.

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