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What is the probability function of $\text{Beta}(1/2, 1/2)$ supposed to look like?

Something like: (ignoring the constant part)

$\theta^{-0.5}(1-\theta)^{-0.5} $ or do we keep the exponents positive even though $0.5-1 = -0.5$?

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This is correct. There's no reason to suddenly make the exponent positive. Here's a graph:

Here's the image!

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