Final Answer:
The exponential family variance functions using \(f\) in place of \(\phi\) and \(m\) in place of \(\mu\) are expressed as \(Var(Y) = f'(\theta)m''(\theta)\).
Step-by-step explanation:
The exponential family variance function is a crucial concept in statistics, particularly when dealing with generalized linear models. In this context, we use \(f\) to represent the cumulant-generating function, which is typically denoted as \(\phi\), and \(m\) to represent the mean function, usually denoted as \(\mu\). The variance function, denoted as \(Var(Y)\), is a fundamental measure of the dispersion or spread of a distribution.
The expression \(Var(Y) = f'(\theta)m''(\theta)\) represents the second derivative of the cumulant-generating function \(f\) with respect to the natural parameter \(\theta\), multiplied by the second derivative of the mean function \(m\) with respect to \(\theta\). This formulation is derived from the fact that the variance of a random variable from the exponential family is determined by the second-order behavior of the cumulant-generating function and mean function.
In mathematical terms, the second derivatives \(f'(\theta)\) and \(m''(\theta)\) capture the curvature or concavity of the respective functions at a given point. Multiplying these curvature measures yields the variance of the distribution. Therefore, the exponential family variance function provides a concise and insightful way to understand the variability in the context of generalized linear models.