Functions of a random variable

(May 9, 2018)

Assume we have a random variable, \(X\), with expected value \(\eta\) and variance \(\sigma^2\).  Often we find ourselves wanting to know the expected value and variance of a function of that random variable, \(f(X)\).  Fortunately there are some workable approximations involving only \(\eta\), \(\sigma^2\) and the derivatives of \(f\).  In both cases we make use of a Taylor-series expansion of \(f(X)\) around \(\eta\):

\[f(X)=\sum_{n=0}^\infty \frac{f^{(n)}(\eta)}{n!}(X-\eta)^n\]

where \(f^{(n)}\) denotes the \(n^{\rm th}\) derivative of \(f\) with respect to \(X\).  For the expected value of \(f(X)\) we then have the following second-order approximation:

\[{\rm E}[f(X)] \approx f(\eta)+\frac{f''(\eta)}{2}\sigma^2\qquad(1)\]

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Tags: GLM, log link, logit link

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