Homework for Module 2
1. Let x have an exponential distribution
p(x|ϑ) = ?
ϑe−ϑx, x ≥ 0
0, otherwise.
(a) Sketch p(x|ϑ) versus x for a fixed value of the parameter ϑ.
(b) Sketch p(x|ϑ) versus ϑ, ϑ 0 for a fixed value of x.
(c) Suppose that n samples x1, . . . , xn are drawn independently according to p(x|ϑ). Show
that the maximum likelihood estimate for ϑ is given by
ϑˆ =
1
1
n
Xn
k=1
xk
.
2. Let x have a uniform distribution
p(x|ϑ) = ? 1
ϑ
, 0 ≤ x ≤ ϑ
0, otherwise.
(a) Sketch p(x|ϑ) versus ϑ for an arbitrary value of x.
(b) Suppose that n samples x1, . . . , xn are drawn independently according to p(x|ϑ). Show
that the maximum likelihood estimate for ϑ is maxk xk.
(c) Find the method of moments estimator for ϑ.
3. Let x be a binary (0, 1) vector with multivariate Bernouli distribution
p
