Let X_1, X_2,...X_n by a sequence of independent and identically distributed random variables with density f_X(x; theta) = {1/theta, if 0 <= x <= theta, and 0, otherwise}. Here, theta > 0 is an unknown parameter.

a) Find the method of moments estimator theta^ for theta (using the first moment)

b) Assuming n is large enough so that we can use the same test as if the data was normally distributed, find the 95% confidence interval for theta, based on theta^. In particular, compute it if n = 100 observations have the same mean X_ = 0.9 and the sample variance S^2_n = 0.3.

c) Suppose we know that theta cannot be larger than 2. How large does the sample size n have to be in order for theta^ to have at least 80% chance (approximately) of lying between theta - 0.1 and theta + 0.1? Here, you do not know the sample variance but you can replace it by the estimate of the true variance sigma^2 = sigma^2(theta), using the value theta = 2. For this, you first have to compute the expression of the variance.