Question 4

A set of data was known to follow the distribution

2x (²³

f(x) = 2e-²₁ x

(a)

(b)

(c)

(d)

x > 0

Demonstrate that the function satisfies the conditions for a valid probability density

function by showing that it integrates to 1 over the entire range of possible values.

(10 marks)

Given a dataset consisting of n observations, x₁,x2,.,, demonstrate that the log-

likelihood function is

n

Σ(2x – 2in 1 - 1

nd - (+)²)

{=1

And thus, show that the Maximum Likelihood Estimator is

λ =

E

(5 marks)

(10 marks)

Assume that you collected a sample of n=10 which was known to follow the

distribution above. The collected sample is 1, 3, 6, 4, 4, 8, 10, 11, 3, 8. Determine the

ML estimate for λ.

(5 marks)