Final answer:
The binomial distribution with parameters (4, 6) does belong to the Exponential family, though the parameters appear to be incorrect. The minimal complete sufficient statistic for the binomial distribution's parameter theta is the total number of successes in the sample.
Step-by-step explanation:
The question is asking whether the binomial distribution with parameters (4, 6) belongs to the Exponential family, and what the minimal complete sufficient statistic for the parameter θ (theta) is.
a) The binomial distribution is indeed part of the Exponential family of distributions. A distribution is in the Exponential family if its probability density function (or probability mass function, in the case of discrete distributions) can be expressed in a certain exponential form.
The binomial distribution, because it arises from Bernoulli trials and has a fixed number of independent trials, fits into this family. However, there seems to be an error in the question as the binomial distribution parameters are usually denoted as (n, p) and not (4, 6). Please verify the parameters for the binomial distribution.
b) For the binomial distribution, the minimal complete sufficient statistic for the parameter θ is the total number of successes observed in the sample. This is because the sum of Bernoulli trials, which each binomial trial is, is sufficient to describe the sample's properties regarding the likelihood of success on any given trial.