(Solution) - The independent random variables X1 Xn have the common distribution where the -(2025 Original AI-Free Solution)

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The independent random variables X1,..., Xn have the common distribution

where the parameters ? and ? are positive.
a. Find a two-dimensional sufficient statistic for (?, ?).
b. Find the MLEs of ? and ?.
c. The length (in millimeters) of cuckoos' eggs found in hedge sparrow nests can be modeled with this distribution. For the data
22.0, 23.9, 20.9, 23.8, 25.0, 24.0, 21.7, 23.8, 22.8, 23.1, 23.1, 23.5, 23.0, 23.0,
find the MLEs of ? and ?.