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Data Mining 625.740 Homework for Module 2 SOLVED

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Applied and Computational Mathematics
Data Mining 625.740
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(x|ϑ) = Y
d
i=1
ϑ
xi
i
(1 − ϑi)
1−xi
,
where ϑ = (ϑ1, . . . , ϑd)
T
is an unknown parameter vector, ϑi being the probability that
xi = 1. Show that the maximum likelihood estimate for ϑ is
ϑˆ =
1
n
Xn
k=1
xk.
4. Let x have a Gamma distribution
p(x|α, β) = (
1
Γ(α)βα x
α−1
e
−x/β, x > 0 and α, β > 0
0, otherwise.
(a) Suppose that n samples x1, . . . , xn are drawn independently according to p(x|α, β). Find
the method of moments estimator for α and β.
(b) Show that the exponential distribution is Γ(1, 1/ϑ).

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