STA2601 Assignment 02 Solutions 2026
UNISA
Module: STA2601 – Applied Statistics II
Assignment 02
Unique Number: 612395
, Question 1
Given the probability density function
𝑓𝑋 (𝑥) = 𝜆(1 − 𝜆)𝑥−1 , 𝑥 > 1
and a random sample 𝑋1 , 𝑋2 , … , 𝑋𝑛 .
Construct the likelihood function
𝑛
𝐿(𝜆) = ∏ 𝜆(1 − 𝜆)𝑥𝑖 −1
𝑖=1
𝐿(𝜆) = 𝜆𝑛 (1 − 𝜆)∑(𝑥𝑖 −1)
𝐿(𝜆) = 𝜆𝑛 (1 − 𝜆)∑𝑥𝑖 −𝑛
Take the natural logarithm
ℓ(𝜆) = ln 𝐿(𝜆)
ℓ(𝜆) = 𝑛ln 𝜆 + (∑𝑥𝑖 − 𝑛)ln(1 − 𝜆)
Differentiate with respect to 𝝀
𝑑ℓ 𝑛 ∑𝑥𝑖 − 𝑛
= −
𝑑𝜆 𝜆 1−𝜆
Set equal to zero:
𝑛 ∑𝑥𝑖 − 𝑛
=
𝜆 1−𝜆
Cross multiply:
𝑛(1 − 𝜆) = 𝜆(∑𝑥𝑖 − 𝑛)
𝑛 = 𝜆∑𝑥𝑖
Solve for 𝝀
𝑛
𝜆̂ =
∑𝑛𝑖=1 𝑥𝑖
UNISA
Module: STA2601 – Applied Statistics II
Assignment 02
Unique Number: 612395
, Question 1
Given the probability density function
𝑓𝑋 (𝑥) = 𝜆(1 − 𝜆)𝑥−1 , 𝑥 > 1
and a random sample 𝑋1 , 𝑋2 , … , 𝑋𝑛 .
Construct the likelihood function
𝑛
𝐿(𝜆) = ∏ 𝜆(1 − 𝜆)𝑥𝑖 −1
𝑖=1
𝐿(𝜆) = 𝜆𝑛 (1 − 𝜆)∑(𝑥𝑖 −1)
𝐿(𝜆) = 𝜆𝑛 (1 − 𝜆)∑𝑥𝑖 −𝑛
Take the natural logarithm
ℓ(𝜆) = ln 𝐿(𝜆)
ℓ(𝜆) = 𝑛ln 𝜆 + (∑𝑥𝑖 − 𝑛)ln(1 − 𝜆)
Differentiate with respect to 𝝀
𝑑ℓ 𝑛 ∑𝑥𝑖 − 𝑛
= −
𝑑𝜆 𝜆 1−𝜆
Set equal to zero:
𝑛 ∑𝑥𝑖 − 𝑛
=
𝜆 1−𝜆
Cross multiply:
𝑛(1 − 𝜆) = 𝜆(∑𝑥𝑖 − 𝑛)
𝑛 = 𝜆∑𝑥𝑖
Solve for 𝝀
𝑛
𝜆̂ =
∑𝑛𝑖=1 𝑥𝑖
