Suppose that X1, . . . , Xn form a random sample from a uniform distribution with the following p.d.f.:

f(x|) = 1/ *I( x 2),where > 0 is the unknown parameter of interest.

(a) Whats the support of _i=1 Xi?

(b) Let Z = min{X1, . . . , Xn} be the minimum of X1, . . . , Xn. Find the CDF of Z, i.e., P(Z z). (Hint: consider P(Z > z), and dont forget to include therange of values that z could take)

(c) Whats the MLE of ?

(d) Heres the observed data, x₁ = 4 (only one data point been collected).Whats the maximum likelihood estimate of ?

(e) Find an estimator for using method of moments.

(f) Whats the mode of f(x|)?

(g) Whats the expectation of X1?

(h) Apply WLLN on the sample mean Xn. State the result.

(i) Let 0 as n . Whats the limiting distribution of f? (Hint: check the support)