2
Q. Consider a random sample X1, X2, ..., Xn from a Gaussian Population with mean 0 and
unknown variance 𝜎 > 0. The parameter of interest is 𝜏 = 1
2𝜎2. To simplify the notation, you can
use 𝑋̿ = ∑ 𝑋𝑖
21
𝑛
𝑛
𝑖=1 .
Solution.
From the above question, then making 𝜎 the subject give;
𝜎2 = 1
2𝜏
The pdf of the Gaussian distribution will be given as;
𝑓(𝑥, 𝜎) = 1
√2𝜋𝜎2 ∗ exp {− 1
2𝜎2 ( 𝑋𝑖 − 0)2} 1
Changing the parameterization yields;
𝑓(𝑥, 𝜏) = √𝜏
√𝜋 exp{−𝜏(𝑋𝑖)2}
a) The Moments Estimator of 𝜏 is given by;
𝜇2
′ = 𝑀2
′ 2
By MGF technique the second moment for the gaussian distribution will be given by;
𝑀𝑥(𝑡) = exp (𝜇𝑡 + 𝜎2𝑡2
2 ) 3
Which on substitution and finding the second derivative yields the second-row population
moments as;
𝜇2
′ = 𝜎2 = 1
2𝜏
So that 𝜏̂ = 1
2𝜎2 which should be equated to the sample raw moments X̿ = ∑ Xi
21
n
n
i=1 .
Since 𝜎2̂ = X̿
Then 𝜏̂ = 1
2X̿ is the MoM estimator.
1 Takeyuki Hida, ‘5. Gaussian Processes’, Stationary Stochastic Processes. (MN-8) (2015).
2 Courtney Taylor, ‘Moment Generating Function for Binomial Distribution’ (2020)
<https://www.thoughtco.com/moment-generating-function-binomial-distribution-3126454> accessed 8
April 2020.
3 ibid.

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