Question

Decide whether the following probability density function is a member of an exponential family
P(x,a)=a^x(1-a)^(1-x)0<a<1,x={0,1}

Answer 1

Solution:

We know that exponential family of distribution is of the form:

$P(x,\theta)=h(x).\exp(\theta^TT(x)-A(\theta))$ ...(i)

Given, $P(x,a)=a^x(1-a)^{1-x}; 0<a<1;x=0,1$

$=\exp[\log (a^x(1-a)^{1-x}) ] \\=\exp[\log a^x+\log(1-a)^{1-x} ] \\=\exp[x\log a+(1-x)\log(1-a) ] \\=\exp[x\log a+\log(1-a)-x\log(1-a) ] \\=\exp[x\log \dfrac{a}{1-a}+\log(1-a) ] \\=\exp[x\theta-\log(1+e^{\theta})] ...(ii)$

where $T(x)=x,\theta=\log \dfrac{a}{1-a},A(\theta)=\log(1+e^{\theta})$

Hence, (ii) matches with (i), given probability density function is a member of an exponential family