The binomial distribution, with distribution function
[mathjax]f_ p(x) = \binom {n}{x}p^ x (1-p)^{n-x}[/mathjax]
can be written as a canonical exponential family, as long as [mathjaxinline]n[/mathjaxinline] is a fixed number. For this problem, plug in [mathjaxinline]n = 1000[/mathjaxinline].
What is the canonical link function [mathjaxinline]g(\mu )[/mathjaxinline]? (With the understanding that [mathjaxinline]\mu = np[/mathjaxinline])
The canonical link function for the binomial distribution is the logit function.
The logit function is given by:
[mathjax]g(\mu) = \log\left(\frac{\mu}{1-\mu}\right)[/mathjax]
In this case, [mathjax]\mu = np[/mathjax], so we have:
[mathjax]g(\mu) = \log\left(\frac{np}{1-np}\right)[/mathjax]
Substituting [mathjax]n = 1000[/mathjax]:
[mathjax]g(\mu) = \log\left(\frac{1000p}{1-1000p}\right)[/mathjax]
In a binomial distribution, the function [mathjaxinline]g(\mu)[/mathjaxinline] is the canonical link function. In this case, [mathjaxinline]\mu = np[/mathjaxinline], where [mathjaxinline]n[/mathjaxinline] is the fixed number and [mathjaxinline]p[/mathjaxinline] is the probability of success.
Therefore, the canonical link function [mathjaxinline]g(\mu)[/mathjaxinline] is simply the identity function, [mathjaxinline]g(\mu) = \mu[/mathjaxinline].