四、(20%) Let X₁, X₂, ..., Xₙ be a random sample from distributions with the following probability density functions: f(x; θ) = 1/2θ^2 x^2 e^-x/θ, 0 &lt; x &lt; ∞, 0 &lt; θ &lt; ∞. Please find the maximum likelihood estimator θ.

五、(25%) To test two method of instruction, 50 students are selected at random for each of two groups. At the end of the instruction period each student is assigned a grade (A, B, C, D, or F) by an evaluating team. The data are recorded as follows: | | Grade | | | | | | | --- | --- | --- | --- | --- | --- | --- | | | A | B | C | D | F | Total | | Group I | 8 | 13 | 16 | 10 | 3 | 50 | | Group II | 4 | 9 | 14 | 16 | 7 | 50 | Please use a chi-square test with α = 0.05 to test the equality of the distributions of grade. The critical value of chi-square distribution for above test is as follows: | r | 2 | 3 | 4 | 5 | 6 | | --- | --- | --- | --- | --- | --- | | χ²₀.₀₅(r) | 5.991 | 7.815 | 9.488 | 11.070 | 12.590 | | χ²₀.₀₂₅(r) | 7.378 | 9.348 | 11.140 | 12.830 | 14.450 | | * P(≥χ²ₐ(r)) = α for X ~ χ²(r). | | | | | |

四、(20%) Let X₁, X₂, ..., Xₙ be a random sample from distributions with the following probability density functions: $$f(x; \theta) = \frac{1}{2\theta^2} x^2 e^{-x/\theta}, 0 < x < \infty, 0 < \theta < \infty.$$ Please find the maximum likelihood estimator $\hat{\theta}$.

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