3. Maximum likelihood estimation

Let x ∈ R be uniformly distributed in the interval [0, θ] where θ is a parameter. That is, the pdf of x is given by

Suppose that n samples D = {x1, . . . , xn} are drawn independently according to fθ(x).

(a) Let fθ(x1, x2, . . . , xn) denote the joint pdf of n independent and identically distributed (i.i.d.) samples drawn according to fθ(x). Express fθ(x1, x2, . . . , xn) as a function of fθ(x1), fθ(x2), . . . , fθ(xn)

(b) We define the maximum likelihood estimate by the value of θ which maximizes the likelihood of having generated the dataset D from the distribution fθ(x). Formally,

Show that the maximum likelihood estimate of θ is max(x1, . . . , xn)