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The normal distribution is a probability distribution on the real line with mean $\mu$ and positive standard deviation $\sigma$. Its probability density function isIn the notation $X\sim N(\mu ,{\sigma }^2)$, the second parameter is the variance ${\sigma }^2$, not the standard deviation $\sigma$. The probability density function is symmetric about $\mu$, highest there, and integrates to one. These formulas and the parameter convention are given in MIT 18.05, Reading 5c, §5: normal distributions.

The normal distribution is a location-scale family: if $Z$ is a standard normal random variable, then $X=\mu +\sigma Z$. Changing $\mu$ moves the center; changing $\sigma$ stretches the horizontal distances from it. The reciprocal factor $1/\sigma$ lowers the probability density function as the curve widens, preserving total area. This is why one standard normal distribution suffices to calculate probabilities for the whole family. ^[ref].

The height $f_X(x)$ is a probability density function value, not $\mathbb{P}(X=x)$. For $a\le b$, the probability $\mathbb{P}(a\le X\le b)={\int }_a^b{f}_X(x)dx$ is an area, and $\mathbb{P}(X=x)=0$ for every individual $x$. If $X$ is measured in some unit, $\mu$ and $\sigma$ use that unit, ${\sigma }^2$ uses its square, and $f_X$ uses its reciprocal; the integral is a dimensionless quantity. A change of measurement unit therefore changes the height without changing the probability. The area interpretation is developed in MIT 18.05, Reading 5b: continuous random variables.

The usefulness of the normal distribution goes beyond its simple shape. Under the central limit theorem, sums of independent and identically distributed random variables with finite positive variance, after centering and scaling, approach the standard normal distribution. Thus approximations by a normal distribution can arise even when the individual random variables have quite different probability distributions. The hypotheses matter: a model of measurement error still needs evidence that this approximation is appropriate. ^[ref]. Exact closure under a linear combination of independent normal random variables provides another reason the family is convenient.

Historically, the curve appeared in a different problem from fitting measurements. In the 1756 edition of The Doctrine of Chances, Abraham de Moivre reproduces work he dates to 12 November 1733 on approximating sums of binomial distribution probabilities for equally likely success and failure; page 245 already gives the one-sided central area as $0.341344$. ^[ref]. In his 1809 Theoria motus, Carl Friedrich Gauss derived a normal distribution for measurement error from a condition on the arithmetic mean, then connected maximizing its likelihood to minimizing the sum of squared errors under that model, the principle behind ordinary least squares. These are distinct contributions: approximating binomial distribution probabilities and justifying an estimation rule. ^[ref].

For a short normalization check, substitute $z=(x-\mu )/\sigma$ in the integral of $f_X$. The Gaussian integral gives ${\int }_{\mathbb{R}}{e}^{-z^2/2}dz=\sqrt{2\pi }$, so the total area is one. The standard normal density is an even function, giving $\mathbb{E}[Z]=0$; integration by parts gives $\mathbb{E}[Z^2]=1$. Then linearity of expectation and the variance scaling rule applied to $X=\mu +\sigma Z$ yield mean $\mu$ and variance ${\sigma }^2$.

Any unbiased linear estimator has the form considered in part (b), and its error is the linear combination of independent normal random variablesThe moment-generating function of a normal distribution therefore gives, for every nonzero real $\theta$,Since the exponential function is strictly increasing and ${\theta }^2{\sigma }^2/2>0$, minimizing this exponential moment is exactly the same as minimizing ${\sum }_i{a}_i^2$. Part (b) shows that the unique minimizer is $a_i=x_i/S_{xx}$. Thus, independently of the sign or magnitude of $\theta$,

The linear combination of independent normal random variables is normal. HenceThis also follows by multiplying the moment-generating functions of the independent and identically distributed random variables.

For every $h\in L^2[0,1]$, the continuous linear functional induced by the inner product givesThis is a normal random variable because it is a linear combination of independent normal random variables. Hence the law of $U$ is a Gaussian measure. Its mean is zero, and independence together with $\mathrm{Var}({\xi }_i)=1/2$ givesThus its covariance operator of a Gaussian measure is

Put $c_i={\int }_D{\phi }_i(x)d{\lambda }_n(x)$. The stated scalar random variable isAs a finite linear combination of independent normal random variables, it is normally distributed. Its mean is zero and its variance is

Finally, orthonormality of the ${\phi }_i$ givesSince $\mathbb{E}{\xi }_i^2=1/2$,