(10%) Let X and Y denote independent random variables with respective probability density function f(x) = 2x, 0 < x < 1, zero elsewhere, and g(y) = 3y<sup>2</sup>, 0 < y < 1, zero elsewhere. Let U = (X, Y) and V = (X, Y). Find the joint probability density function of U and V.

(e) (5%) Suppose a random sample of size 49 is collected as follows:0.06 0.06 0.09 0.15 0.17 0.17 0.18 0.21 0.23 0.29 0.29 0.31 0.34 0.34 0.370.41 0.41 0.42 0.44 0.45 0.46 0.47 0.47 0.49 0.49 0.49 0.50 0.53 0.56 0.570.57 0.58 0.58 0.59 0.59 0.60 0.63 0.64 0.66 0.68 0.68 0.71 0.74 0.78 0.810.85 0.85 0.91 0.97.We obtain $\sum{i=1}^{49} xi = 23.84$ and $\sum{i=1}^{49} \log(xi) = -43.052$. Find $\hat{\xi}{0.6}$ and $\hat{\xi}{0.6}$.

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