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).

625.740

# Data Mining 625.740 Homework for Module 2 SOLVED

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