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I am working on a probability theory question related to coalescent theory in population genetics, and the following distribution occurred in my derivation. For three values $a, b, c$, each between 0 and 1, let

$$ X = \begin{cases}1 \text{ with probability }\frac{a}{a+b+c} \\ 0 \text{ with probability } \frac{b}{a+b+c}, \\ -1 \text{ with probability } \frac{c}{a+b+c}. \end{cases} $$

Is there a known name for this class of distributions that I can refer to when writing about it?

Further, for a certain choice of $a, b, c$ I get

$$ \mathbb{E}X = \tanh(\alpha -\beta), $$ where $\alpha, \beta > 0$ are parameters that have a biological interpretation. I fell like I am not the first person to encounter these expressions in a probability context. Have you seen these anywhere else?

Thomas Andrews
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Egor Lappo
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    You have a general discrete distribution with support ${-1,0,1}$: if the values $-1,0,1$ have respective probabilities $p_{-1}, p_0, p_{1}$, then you can choose any positive $k\ge \max\left(p_{-1}, p_0, p_{1}\right)$ and let $a=\frac{p_{1}}k$, $b=\frac{p_{0}}k$, $c=\frac{p_{-1}}k$. Its expectation is $\frac{a-c}{a+b+c} = p_{1}-p_{-1}$. – Henry Oct 02 '24 at 00:39
  • This is not a direct name but $X$ has the distribution of $Y.(-1,0,1)$ where $Y=(y_1,y_2,y_3)$ has a multinomial distribution with parameters $1$ and $(a/(a+b+c),b/(a+b+c),c/(a+b+c)$. – JimB Oct 02 '24 at 02:27

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