Which of the following are examples of one-parameter canonical exponential families for nonzero \theta?
(Select all that apply.)
Normal(\mu ,1)
Poisson(\lambda )
Uniform([0,a])
Bernoulli(p)
Binomial(1000,p)
Exponential(\lambda )
Poisson(\lambda )
Bernoulli(p)
Exponential(\lambda )
To determine which of the given options are examples of one-parameter canonical exponential families for a nonzero value of θ, let's analyze each case:
1. Normal(μ,1):
The normal distribution is not a one-parameter canonical exponential family because it depends on two parameters (mean μ and variance σ^2).
2. Poisson(λ):
The Poisson distribution is a one-parameter exponential family, and it is canonical because it is in the exponential form.
3. Uniform([0,a]):
The uniform distribution is not an exponential family because it does not have an exponential form.
4. Bernoulli(p):
The Bernoulli distribution is a one-parameter exponential family, and it is canonical because it is in the exponential form.
5. Binomial(1000,p):
The binomial distribution is not a one-parameter canonical exponential family because it has two parameters (n and p).
6. Exponential(λ):
The exponential distribution is a one-parameter exponential family, and it is canonical because it is in the exponential form.
Therefore, the options that are examples of one-parameter canonical exponential families for a nonzero θ are:
- Poisson(λ)
- Bernoulli(p)
- Exponential(λ)