Normal distribution
Estimation of parameters

It is often the case that we do not know the parameters of the normal distribution, but instead want to estimate them. That is, having a sample (x 1, …, x n) {\textstyle (x_{1},\ldots,x_{n})} from a normal N (μ, σ 2) {\textstyle {\mathcal {N}}(\mu,\sigma ^{2})} population we would like to learn the approximate values of parameters μ {\textstyle \mu } and σ 2 {\textstyle \sigma ^{2}}. The standard approach to this problem is the maximum likelihood method, which requires maximization of the log-likelihood function: ln L (μ, σ 2) = ∑ i = 1 n ln f (x i ∣ μ, σ 2) = − n 2 ln (2 π) − n 2 ln σ 2 − 1 2 σ 2 ∑ i = 1 n (x i − μ) 2. {\displaystyle \ln {\mathcal {L}}(\mu,\sigma ^{2})=\sum _{i=1}^{n}\ln f(x_{i}\mid \mu,\sigma ^{2})=-{\frac {n}{2}}\ln(2\pi)-{\frac {n}{2}}\ln \sigma ^{2}-{\frac {1}{2\sigma ^{2}}}\sum _{i=1}^{n}(x_{i}-\mu)^{2}.} Taking derivatives with respect to μ {\textstyle \mu } and σ 2 {\textstyle \sigma ^{2}} and solving the resulting system of first order conditions yields the maximum likelihood estimates: μ ^ = x ¯ ≡ 1 n ∑ i = 1 n x i, σ ^ 2 = 1 n ∑ i = 1 n (x i − x ¯) 2. {\displaystyle {\hat {\mu }}={\overline {x}}\equiv {\frac {1}{n}}\sum _{i=1}^{n}x_{i},\qquad {\hat {\sigma }}^{2}={\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}.}

Then ln L (μ ^, σ ^ 2) {\textstyle \ln {\mathcal {L}}({\hat {\mu }},{\hat {\sigma }}^{2})} is as follows:

ln L (μ ^, σ ^ 2) = (− n / 2) {\displaystyle \ln {\mathcal {L}}({\hat {\mu }},{\hat {\sigma }}^{2})=(-n/2)}
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