statistics?

Question

Let X1, · · · ,Xn be a random sample from uniform distribution with density
f(x) =1/θ, 0 < x < θ, θ > 0
(a) Find the maximum likelihood estimator θ ˆ of θ .
(b) Compute the cumulative distribution function (c.d.f) of θˆ  .
(c) Use to construct the 95% equal-tail confidence interval for θ .

Answer 1

Since it is a uniform distribution over the range [0, theta], the chances of choosing any one number in the range are the same as choosing any other number.

a) Suppose you chose a large number of numbers from this distribution. What would you expect the average to be?

b) What percentage of those numbers would you expect to be less than theta/4? less than theta/2? less than (3/4)theta?

c) If the two tails are equal, they each contain 2.5%. So the lower bound of the interval is the number X such that 2.5% of the random numbers you chose can be expected to be less than X. The upper bound, Y, is that number such that 97.5% of the numbers can be expected to be less than Y.

HTH.