Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 38 / 4 / a / Solution

The ratio of expected process variance to variance of hypothetical means is $K=s^2/v=(\alpha +1)/(k-2)$, and the credibility factor is $Z=n/(n+K)$. Holding $\alpha ,\lambda$ fixed, an increase in $k$ decreases $K$. Hence the credibility factor increases, approaching one as $k$ tends to infinity. Extending the formula to real $k>2$ makes the sign explicit:For the integer values in the model this establishes the same strict monotonicity. The collective claim expected value $m=\alpha /[\lambda (k-1)]$ also changes, but the conclusion concerns the relative weight of experience: conditional noise declines faster than the variation of the risk means.