 Solution

Both candidate models fit one parameter, but their parameter symbols have different meanings. For the exponential distribution in model B,Its second derivative is $-N/{\theta }_B^2<0$, so this is its maximum-likelihood estimate. The maximized log-likelihoods, including their normalization terms, areUse the corrected Akaike information criterion because $N=4$ is small relative to the number $k=1$ of fitted parameters:The correction is $2$ for each model, givingPrefer model A, whose corrected Akaike information criterion is smaller by $7.67252$. The ordinary Akaike information criterion also gives that ranking because the two models receive equal penalties. This compares the two proposed models; it does not establish that either is a correct description of the process. The usual corrected Akaike information criterion formula is a small-sample model-selection convention, rather than an exact finite-sample guarantee for these density families.