Continuous binomial distribution

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A random variable ${\displaystyle X}$ is said to follow a continuous binomial (cobin) distribution with natural parameter ${\displaystyle \theta }$ and inverse dispersion ${\displaystyle \lambda \in \{1,2,\dots \}}$, written ${\displaystyle X\sim \mathrm {cobin} (\theta ,\lambda ^{-1})}$, if it has density on ${\displaystyle [0,1]}$ given by

${\displaystyle f(x;\theta ,\lambda )=h(x;\lambda )\exp \!{\big (}\lambda \theta x-\lambda B(\theta ){\big )},\qquad 0\leq x\leq 1,}$

where the log-partition function is

${\displaystyle B(\theta )={\begin{cases}\log \!\left({\frac {e^{\theta }-1}{\theta }}\right),&\theta \neq 0,\\0,&\theta =0,\end{cases}}}$

and the base measure ${\displaystyle h(x;\lambda )}$ is

${\displaystyle h(x;\lambda )={\frac {\lambda }{(\lambda -1)!}}\sum _{k=0}^{\lambda }(-1)^{k}{\lambda \choose k}\,\max(0,\lambda x-k)^{\lambda -1},}$

with ${\displaystyle h(x;1)=1}$. The function ${\displaystyle h(x;\lambda )/\lambda }$ coincides with the probability density function of the Irwin–Hall distribution with parameter ${\displaystyle n=\lambda }$, evaluated at ${\displaystyle \lambda x}$.

When ${\displaystyle \lambda }$ is fixed, the cobin distribution belongs to a one-parameter natural exponential family in ${\displaystyle \theta }$.

• Continuous Bernoulli distribution: when ${\displaystyle \lambda =1}$, the cobin distribution reduces to the continuous Bernoulli distribution.

• Bates distribution: when ${\displaystyle \theta =0}$, the density reduces to ${\displaystyle h(x;\lambda )}$, corresponding to the distribution of the mean of ${\displaystyle \lambda }$ independent ${\displaystyle \mathrm {Uniform} (0,1)}$ random variables (equivalently, a scaled Irwin–Hall distribution or Bates distribution).

• Uniform distribution: when ${\displaystyle \lambda =1}$ and ${\displaystyle \theta =0}$, the distribution reduces to the continuous uniform distribution on ${\displaystyle [0,1]}$.

• If ${\displaystyle X_{1},\dots ,X_{\lambda }}$ are independent and identically distributed continuous Bernoulli random variables with common natural parameter ${\displaystyle \theta }$, then

${\displaystyle {\bar {X}}={\frac {1}{\lambda }}\sum _{i=1}^{\lambda }X_{i}\sim \mathrm {cobin} (\theta ,\lambda ^{-1}).}$

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Mean and variance

The mean and variance of ${\displaystyle X\sim \mathrm {cobin} (\theta ,\lambda ^{-1})}$ can be expressed in terms of derivatives of ${\displaystyle B(\theta )}$:

• ${\displaystyle \operatorname {E} (X)=B'(\theta )={\frac {e^{\theta }}{e^{\theta }-1}}-{\frac {1}{\theta }}}$, for ${\displaystyle \theta \neq 0}$.
• ${\displaystyle \operatorname {Var} (X)={\frac {1}{\lambda }}B''(\theta )={\frac {1}{\lambda }}\left({\frac {1}{\theta ^{2}}}-{\frac {e^{\theta }}{(e^{\theta }-1)^{2}}}\right)}$, for ${\displaystyle \theta \neq 0}$.

If ${\displaystyle \theta =0}$, then ${\displaystyle \operatorname {E} (X)={\frac {1}{2}}}$ and ${\displaystyle \operatorname {Var} (X)={\frac {1}{12\lambda }}}$.

Sufficient statistic for the mean

If ${\displaystyle X_{1},\ldots ,X_{n}}$ are independent and identically distributed continuous binomial random variables with common natural parameter ${\displaystyle \theta }$ and fixed inverse dispersion parameter ${\displaystyle \lambda }$, then the sample mean

${\displaystyle {\bar {X}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}}$

is a sufficient statistic for ${\displaystyle \theta }$.

This is in contrast with the beta distribution: under a mean–precision parameterisation ${\displaystyle X_{i}\sim \mathrm {Beta} (\mu \phi ,(1-\mu )\phi )}$ with fixed ${\displaystyle \phi }$, a sufficient statistic for the mean ${\displaystyle \mu }$ is

${\displaystyle \sum _{i=1}^{n}\log \!\left({\frac {X_{i}}{1-X_{i}}}\right),}$

not the sample mean ${\displaystyle {\bar {X}}}$.