Consider a Normal population with mean
𝜇
and variance
𝜎
2

Question

Consider a Normal population with mean 
𝜇
 and variance 
𝜎
2
<
∞
. We collect a sample of 
𝑇
=
3
 independent and identically distributed observations
𝑥
=
(
4
,
10
,
7
)
In class we have shown that the standardized log-likelihood of the sample for this case is
1
𝑇
log
(
𝐿
(
𝑥
,
𝜇
,
𝜎
2
)
)
=
−
1
2
log
(
2
𝜋
)
−
1
2
log
(
𝜎
2
)
−
1
𝑇
∑
𝑡
=
1
𝑇
(
𝑥
𝑡
−
𝜇
)
2
2
𝜎
2
.
Then the Maximum Likelihood estimate of the variance 
𝜎
2
 is:

BlackTom AI · Accepted Answer

For a normal distribution with unknown mean and variance, the MLE for σ^2 equals the average of squared deviations from the MLE of μ (the sample mean).

1) Relevant knowledge points and concepts
- For independent normal observations with unknown mean μ and variance σ^2, the log-likelihood (per sample) yields MLEs: μ̂ = sample mean x̄ and σ̂^2 = (1/T) Σ (x_t − μ̂)^2. Note the MLE uses denominator T (not T−1 used for the unbiased estimator).

2) Complete reasoning (derivation)
Starting from the given standardized log-likelihood per observation:
l(μ,σ^2) = −(1/2) log(2π) − (1/2) log(σ^2) − [1/(2σ^2 T)] Σ (x_t − μ)^2.
Differentiate w.r.t. μ:
∂l/∂μ = −[1/(2σ^2 T)] * 2 Σ (x_t − μ) = −[1/(σ^2 T)] Σ (x_t − μ).
Setting ∂l/∂μ = 0 gives Σ (x_t − μ̂) = 0 ⇒ μ̂ = x̄.
Differentiate w.r.t. σ^2:
l = const − (1/2) log σ^2 − S/(2σ^2), with S = (1/T) Σ (x_t − μ)^2.
∂l/∂σ^2 = −1/(2σ^2) + S/(2σ^4) = 0 ⇒ S = σ̂^2.
Thus σ̂^2 = (1/T) Σ (x_t − μ̂)^2.

3) Calculations with intermediate steps
Data x = (4, 10, 7), T = 3.
Compute sample mean: x̄ = (4 + 10 + 7)/3 = 21/3 = 7.
Squared deviations: (4−7)^2 = 9, (10−7)^2 = 9, (7−7)^2 = 0. Sum = 9 + 9 + 0 = 18.
σ̂^2 = (1/3) * 18 = 6.

4) Why the correct answer is correct
The MLE for variance uses the average squared deviation from the MLE mean; applying that formula to these observations yields 6 exactly, so σ̂^2 = 6 is the MLE.

5) Why other options are incorrect
- σ̂^2 = 9: this is the unbiased sample variance using denominator T−1: 18/2 = 9, not the MLE.
- σ̂^2 = 15: does not match any correct variance formula for these data (sum of squares or mean squares do not produce 15).
- σ̂^2 = 8: arbitrary and not produced by any correct estimator here.
- "There is not enough information": incorrect because the full sample is provided and the MLE is computable.

6) Conclusion
The Maximum Likelihood estimate of the variance is σ̂^2 = 6, matching the provided answer.

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