Exponential family

Exponential Family 

Exponential family is a specific probability distribution of a certain form. In statistics, there are some properties that distributions of exponential family have in common. and lots of familiar distributions including exponential distribution, Bernoulli, Beta, Gamma, Poisson, Negative Binomial, Geometric, chi-squared, normal distribution are the form of exponential family. 

<specific form of an exponential family>

If any distribution can be expressed as this form, the distribution can be called 'exponential family'

In the form, X is the random variable and θ is a parameter of distributions. 

If the range of the random variable X does not rely on parameter θ, the condition for being an exponential family is satisfied. 

< Exponential distribution >

Exponential distribution is representative of the exponential family. 

< Bernoulli distribution >

Bernoulli distribution is a form of the exponential family. 

The parameter of Bernoulli distribution is 'p' and the range of random variable X is {0,1} does not rely on the parameter p. 

So Bernoulli distribution is an exponential family of parameter p.

< Poisson distribution >

Poisson distribution is a form of the exponential family. 

The parameter of Poisson distribution is ''lambda" and the range of random variable X is {0,1,2,3,...} does not rely on the parameter lambda. 

So Bernoulli distribution is an exponential family of parameter lambda.

< Normal distribution >

Normal distribution is a form of the exponential family. 

The parameter of Normal distribution is mean and variance and the range of random variable X is {0,1,2,3,...} does not rely on the parameter p. 

So Normal distribution is an exponential family of parameter p.

< Beta distribution >

Beta distribution is a form of the exponential family. 

The parameter of Beta distribution is alpha and Beta and the range of random variable X is [0,1] does not rely on the parameter Alpha, Beta.

So Beta distribution is an exponential family of parameter alpha and beta.

< Gamma distribution >

Gamma distribution is a form of the exponential family. 

The parameter of Gamma is mean and variance and the range of random variable X is {0,1,2,3,...} does not rely on the parameter p. 

So Gamma distribution is an exponential family of parameter p.

< Binomial distribution >

Binomial distribution is a form of the exponential family. 

The parameter of Binomial is alpha and beta and the range of random variable X is (0, ∞) does not rely on the parameter alpha and beta. 

So Binomial distribution is an exponential family of parameter alpha and beta.

< Chi-squared distribution >

Chi-squared is a form of the exponential family. 

The parameter of Chi-squared is alpha and beta and the range of random variable X is (0, ∞) does not rely on the parameter n.

So Chi-squared distribution is an exponential family of parameter n.

< Geometric  distribution >

Geometric distribution is a form of the exponential family. 

The parameter of Geometric is alpha and beta and the range of random variable X is {1,2,3...} does not rely on the parameter p. 

So Geometric distribution is an exponential family of parameter n.

< Negative Binomial distribution >

Negative Binomial distribution is a form of the exponential family. 

The parameter of Negative Binomial is r and p and the range of random variable X is {r, r+1, r+2,...} does not rely on the parameter r and p.

So Negative Binomial distribution is an exponential family of parameter r and p. 

<Uniform distribution - not an exponential family>

Uniform distribution is not a form of the exponential family. 

As you know, range of random variable of U(a,b) for f(x)>0 is a<X<b where a and b are parameters of the uniform distribution. So Uniform distribution is not an exponential family because the range of random variable rely on parameters a and b.