The binomial distribution, with distribution function
f_ p(x) = \binom {n}{x}p^ x (1-p)^{n-x}
can be written as a canonical exponential family, as long as n is a fixed number. For this problem, plug in n = 1000.
What is the canonical link function g(\mu )? (With the understanding that \mu = np)
The canonical link function for the binomial distribution with parameter \mu = np is the logit function, given by:
g(\mu) = \log\left(\frac{\mu}{1-\mu}\right)
In this case, since \mu = np, we have:
g(\mu) = \log\left(\frac{np}{1-np}\right)
The canonical link function for the binomial distribution is given by:
g(μ) = ln(μ / (1 - μ))
In this case, since μ = np, substituting this value into the canonical link function we have:
g(np) = ln(np / (1 - np))
However, since np is denoted as μ, we can rewrite the equation as:
g(μ) = ln(μ / (1 - μ))