= Solution
Both candidate models fit one parameter, but their parameter symbols have different meanings. For the <exponential distribution> in model B,
$$
\ell_B(\theta_B)=N\log\theta_B-\theta_B\sum_i x_i,\qquad
\widehat\theta_B=\frac{N}{\sum_i x_i}=\frac{10}{7}.
$$
Its second derivative is $-N/\theta_B^2<0$, so this is its <maximum-likelihood estimate>. The maximized <log-likelihoods>, including their normalization terms, are
$$
\ell_A(\widehat\theta_A)=1.262960000,\qquad
\ell_B(\widehat\theta_B)=4\log(10/7)-4=-2.573300224.
$$
Use the <corrected Akaike information criterion> because $N=4$ is small relative to the number $k=1$ of fitted parameters:
$$
\operatorname{AICc}=-2\ell(\widehat\theta)+2k+\frac{2k(k+1)}{N-k-1}.
$$
The correction is $2$ for each model, giving
$$
\boxed{\operatorname{AICc}_A=1.47408,\qquad \operatorname{AICc}_B=9.14660}.
$$
\b[Prefer 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.
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