Chapter 6.2, Problem 23E

Let X represent the error in making a measurement of a physical characteristic or property (e.g., the boiling point of a particular liquid). It is often reasonable to assume that E(X) = 0 and that X has a normal distribution. Thus the pdf of any particular measurement error is

$\begin{array}{cccc}f(x;\theta )& =& \frac{1}{\sqrt{2\pi \theta }}{e}^{-x^2/2\theta }& -\infty \text{ < }x\text{ < }\infty \end{array}$

(where we have used θ in place of σ²). Now suppose that n independent measurements are made, resulting in measurement errors X₁ = x₁, X₂ = x₂,..., X_n = x_n. Obtain the mle of θ.