Normal distribution
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Template:Probability distribution
In probability theory, the normal (or Gaussian) distribution is a very commonly occurring continuous probability distribution—a function that tells the probability that any real observation will fall between any two real limits or real numbers, as the curve approaches zero on either side. Normal distributions are extremely important in statistics and are often used in the natural and social sciences for real-valued random variables whose distributions are not known.^{[1]}^{[2]}
The normal distribution is immensely useful because of the central limit theorem, which states that, under mild conditions, the mean of many random variables independently drawn from the same distribution is distributed approximately normally, irrespective of the form of the original distribution: physical quantities that are expected to be the sum of many independent processes (such as measurement errors) often have a distribution very close to the normal.^{[3]} Moreover, many results and methods (such as propagation of uncertainty and least squares parameter fitting) can be derived analytically in explicit form when the relevant variables are normally distributed.
The Gaussian distribution is sometimes informally called the bell curve. However, many other distributions are bell-shaped (such as Cauchy's, Student's, and logistic). The terms Gaussian function and Gaussian bell curve are also ambiguous because they sometimes refer to multiples of the normal distribution that cannot be directly interpreted in terms of probabilities.
A normal distribution is:
The parameter in this definition is the mean or expectation of the distribution (and also its median and mode). The parameter is its standard deviation; its variance is therefore . A random variable with a Gaussian distribution is said to be normally distributed and is called a normal deviate.
If and , the distribution is called the standard normal distribution or the unit normal distribution denoted by and a random variable with that distribution is a standard normal deviate.
The normal distribution is the only absolutely continuous distribution all of whose cumulants beyond the first two (i.e., other than the mean and variance) are zero. It is also the continuous distribution with the maximum entropy for a given mean and variance.^{[4]}^{[5]}
The normal distribution is a subclass of the elliptical distributions. The normal distribution is symmetric about its mean, and is non-zero over the entire real line. As such it may not be a suitable model for variables that are inherently positive or strongly skewed, such as the weight of a person or the price of a share. Such variables may be better described by other distributions, such as the log-normal distribution or the Pareto distribution.
The value of the normal distribution is practically zero when the value x lies more than a few standard deviations away from the mean. Therefore, it may not be an appropriate model when one expects a significant fraction of outliers—values that lie many standard deviations away from the mean — and least squares and other statistical inference methods that are optimal for normally distributed variables often become highly unreliable when applied to such data. In those cases, a more heavy-tailed distribution should be assumed and the appropriate robust statistical inference methods applied.
The Gaussian distribution belongs to the family of stable distributions which are the attractors of sums of independent, identically distributed (i.i.d.) distributions whether or not the mean or variance is finite. Except for the Gaussian which is a limiting case, all stable distributions have heavy tails and infinite variance.
Definition
Standard normal distribution
The simplest case of a normal distribution is known as the standard normal distribution. This is a special case where μ=0 and σ=1, and it is described by this probability density function:
The factor in this expression ensures that the total area under the curve ϕ(x) is equal to one.^{[6]} The Template:Frac2 in the exponent ensures that the distribution has unit variance (and therefore also unit standard deviation). This function is symmetric around x=0, where it attains its maximum value ; and has inflection points at +1 and −1.
Authors may differ also on which normal distribution should be called the "standard" one. Gauss himself defined the standard normal as having variance σ^{2} = Template:Frac2, that is
Stigler^{[7]} goes even further, defining the standard normal with variance σ^{2} = Template:Frac2 :
General normal distribution
Any normal distribution is a version of the standard normal distribution whose domain has been stretched by a factor σ (the standard deviation) and then translated by μ (the mean value):
The probability density must be scaled by so that the integral is still 1.
If Z is a standard normal deviate, then X = Zσ + μ will have a normal distribution with expected value μ and standard deviation σ. Conversely, if X is a general normal deviate, then Z = (X − μ)/σ will have a standard normal distribution.
Every normal distribution is the exponential of a quadratic function:
where a is negative and c is . In this form, the mean value μ is −b/(2a), and the variance σ^{2} is −1/(2a). For the standard normal distribution, a is −1/2, b is zero, and c is .
Notation
The standard Gaussian distribution (with zero mean and unit variance) is often denoted with the Greek letter ϕ (phi).^{[8]} The alternative form of the Greek phi letter, φ, is also used quite often.
The normal distribution is also often denoted by N(μ, σ^{2}).^{[9]} Thus when a random variable X is distributed normally with mean μ and variance σ^{2}, we write
Alternative parameterizations
Some authors advocate using the precision τ as the parameter defining the width of the distribution, instead of the deviation σ or the variance σ^{2}. The precision is normally defined as the reciprocal of the variance, 1/σ^{2}.^{[10]} The formula for the distribution then becomes
This choice is claimed to have advantages in numerical computations when σ is very close to zero and simplify formulas in some contexts, such as in the Bayesian inference of variables with multivariate normal distribution.
Also the reciprocal of the standard deviation might be defined as the precision and the expression of the normal distribution becomes
According to Stigler, this formulation is advantageous because of a much simpler and easier-to-remember formula, the fact that the pdf has unit height at zero, and simple approximate formulas for the quantiles of the distribution.
Properties
Symmetries and derivatives
The normal distribution f(x), with any mean μ and any positive deviation σ, has the following properties:
- It is symmetric around the point x = μ, which is at the same time the mode, the median and the mean of the distribution.^{[11]}
- It is unimodal: its first derivative is positive for x < μ, negative for x > μ, and zero only at x = μ.
- Its density has two inflection points (where the second derivative of f is zero and changes sign), located one standard deviation away from the mean, namely at x = μ − σ and x = μ + σ.^{[11]}
- Its density is log-concave.^{[11]}
- Its density is infinitely differentiable, indeed supersmooth of order 2.^{[12]}
- Its second derivative f′′(x) is equal to its derivative with respect to its variance σ^{2}
Furthermore, the density ϕ of the standard normal distribution (with μ = 0 and σ = 1) also has the following properties:
- Its first derivative ϕ′(x) is −xϕ(x).
- Its second derivative ϕ′′(x) is (x^{2} − 1)ϕ(x)
- More generally, its n-th derivative ϕ^{(n)}(x) is (−1)^{n}H_{n}(x)ϕ(x), where H_{n} is the Hermite polynomial of order n.^{[13]}
- It satisfies the differential equation
- or
Moments
{{#invoke:see also|seealso}} The plain and absolute moments of a variable X are the expected values of X^{p} and |X|^{p},respectively. If the expected value μ of X is zero, these parameters are called central moments. Usually we are interested only in moments with integer order p.
If X has a normal distribution, these moments exist and are finite for any p whose real part is greater than −1. For any non-negative integer p, the plain central moments are
Here n!! denotes the double factorial, that is, the product of every number from n to 1 that has the same parity as n.
The central absolute moments coincide with plain moments for all even orders, but are nonzero for odd orders. For any non-negative integer p,
The last formula is valid also for any non-integer p > −1. When the mean μ is not zero, the plain and absolute moments can be expressed in terms of confluent hypergeometric functions _{1}F_{1} and U.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}^{[citation needed]} }}
These expressions remain valid even if p is not integer. See also generalized Hermite polynomials.
Order | Non-central moment | Central moment |
---|---|---|
1 | μ | 0 |
2 | μ^{2} + σ^{2} | σ^{ 2} |
3 | μ^{3} + 3μσ^{2} | 0 |
4 | μ^{4} + 6μ^{2}σ^{2} + 3σ^{4} | 3σ^{ 4} |
5 | μ^{5} + 10μ^{3}σ^{2} + 15μσ^{4} | 0 |
6 | μ^{6} + 15μ^{4}σ^{2} + 45μ^{2}σ^{4} + 15σ^{6} | 15σ^{ 6} |
7 | μ^{7} + 21μ^{5}σ^{2} + 105μ^{3}σ^{4} + 105μσ^{6} | 0 |
8 | μ^{8} + 28μ^{6}σ^{2} + 210μ^{4}σ^{4} + 420μ^{2}σ^{6} + 105σ^{8} | 105σ^{ 8} |
Fourier transform and characteristic function
The Fourier transform of a normal distribution f with mean μ and deviation σ is^{[14]}
where i is the imaginary unit. If the mean μ is zero, the first factor is 1, and the Fourier transform is also a normal distribution on the frequency domain, with mean 0 and standard deviation 1/σ. In particular, the standard normal distribution ϕ (with μ=0 and σ=1) is an eigenfunction of the Fourier transform.
In probability theory, the Fourier transform of the probability distribution of a real-valued random variable X is called the characteristic function of that variable, and can be defined as the expected value of e^{i tX}, as a function of the real variable t (the frequency parameter of the Fourier transform). This definition can be analytically extended to a complex-value parameter t.^{[15]}
Moment and cumulant generating functions
The moment generating function of a real random variable X is the expected value of e^{tX}, as a function of the real parameter t. For a normal distribution with mean μ and deviation σ, the moment generating function exists and is equal to
The cumulant generating function is the logarithm of the moment generating function, namely
Since this is a quadratic polynomial in t, only the first two cumulants are nonzero, namely the mean μ and the variance σ^{2}.
Cumulative distribution function
The cumulative distribution function (CDF) of the standard normal distribution, usually denoted with the capital Greek letter (phi), is the integral
In statistics one often uses the related error function, or erf(x), defined as the probability of a random variable with normal distribution of mean 0 and variance 1/2 falling in the range ; that is
These integrals cannot be expressed in terms of elementary functions, and are often said to be special functions *. They are closely related, namely
For a generic normal distribution f with mean μ and deviation σ, the cumulative distribution function is
The complement of the standard normal CDF, , is often called the Q-function, especially in engineering texts.^{[16]}^{[17]} It gives the probability that the value of a standard normal random variable X will exceed x. Other definitions of the Q-function, all of which are simple transformations of , are also used occasionally.^{[18]}
The graph of the standard normal CDF has 2-fold rotational symmetry around the point (0,1/2); that is, . Its antiderivative (indefinite integral) is .
- The cumulative distribution function (CDF) of the standard normal distribution can be expanded by Integration by parts into a series:
where denotes the double factorial. Example of Pascal function to calculate CDF (sum of first 100 elements)
function CDF(x:extended):extended;
var value,sum:extended;
i:integer;
begin
sum:=x;
value:=x;
for i:=1 to 100 do
begin
value:=(value*x*x/(2*i+1));
sum:=sum+value;
end;
result:=0.5+(sum/sqrt(2*pi))*exp(-(x*x)/2);
end;
Standard deviation and tolerance intervals
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About 68% of values drawn from a normal distribution are within one standard deviation σ away from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations. This fact is known as the 68-95-99.7 (empirical) rule, or the 3-sigma rule.
More precisely, the probability that a normal deviate lies in the range μ − nσ and μ + nσ is given by
To 12 decimal places, the values for n = 1, 2, …, 6 are:^{[19]}
n | F(μ+nσ) − F(μ − nσ) | i.e. 1 minus … | or 1 in … | OEIS |
---|---|---|---|---|
1 | Template:Val | Template:Val | Template:Val | A178647 |
2 | Template:Val | Template:Val | Template:Val | A110894 |
3 | Template:Val | Template:Val | Template:Val | |
4 | Template:Val | Template:Val | Template:Val | |
5 | Template:Val | Template:Val | Template:Val | |
6 | Template:Val | Template:Val | Template:Val |
Quantile function
The quantile function of a distribution is the inverse of the cumulative distribution function. The quantile function of the standard normal distribution is called the probit function, and can be expressed in terms of the inverse error function:
For a normal random variable with mean μ and variance σ^{2}, the quantile function is
The quantile of the standard normal distribution is commonly denoted as z_{p}. These values are used in hypothesis testing, construction of confidence intervals and Q-Q plots. A normal random variable X will exceed μ + σz_{p} with probability 1−p; and will lie outside the interval μ ± σz_{p} with probability 2(1−p). In particular, the quantile z_{0.975} is 1.96; therefore a normal random variable will lie outside the interval μ ± 1.96σ in only 5% of cases.
The following table gives the multiple n of σ such that X will lie in the range μ ± nσ with a specified probability p. These values are useful to determine tolerance interval for sample averages and other statistical estimators with normal (or asymptotically normal) distributions:^{[20]}
F(μ + nσ) − F(μ − nσ) | n | F(μ + nσ) − F(μ − nσ) | n | |
---|---|---|---|---|
0.80 | Template:Val | 0.999 | Template:Val | |
0.90 | Template:Val | 0.9999 | Template:Val | |
0.95 | Template:Val | 0.99999 | Template:Val | |
0.98 | Template:Val | 0.999999 | Template:Val | |
0.99 | Template:Val | 0.9999999 | Template:Val | |
0.995 | Template:Val | 0.99999999 | Template:Val | |
0.998 | Template:Val | 0.999999999 | Template:Val |
Zero-variance limit
In the limit when σ tends to zero, the probability density f(x) eventually tends to zero at any x ≠ μ, but grows without limit if x = μ, while its integral remains equal to 1. Therefore, the normal distribution cannot be defined as an ordinary function when σ = 0.
However, one can define the normal distribution with zero variance as a generalized function; specifically, as Dirac's "delta function" δ translated by the mean μ, that is f(x) = δ(x−μ). Its CDF is then the Heaviside step function translated by the mean μ, namely
Central limit theorem
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The central limit theorem states that under certain (fairly common) conditions, the sum of many random variables will have an approximately normal distribution. More specifically, where X_{1}, …, X_{n} are independent and identically distributed random variables with the same arbitrary distribution, zero mean, and variance σ^{2}; and Z is their mean scaled by
Then, as n increases, the probability distribution of Z will tend to the normal distribution with zero mean and variance σ^{2}.
The theorem can be extended to variables X_{i} that are not independent and/or not identically distributed if certain constraints are placed on the degree of dependence and the moments of the distributions.
Many test statistics, scores, and estimators encountered in practice contain sums of certain random variables in them, and even more estimators can be represented as sums of random variables through the use of influence functions. The central limit theorem implies that those statistical parameters will have asymptotically normal distributions.
The central limit theorem also implies that certain distributions can be approximated by the normal distribution, for example:
- The binomial distribution B(n, p) is approximately normal with mean np and variance np(1−p) for large n and for p not too close to zero or one.
- The Poisson distribution with parameter λ is approximately normal with mean λ and variance λ, for large values of λ.^{[21]}
- The chi-squared distribution χ^{2}(k) is approximately normal with mean k and variance 2k, for large k.
- The Student's t-distribution t(ν) is approximately normal with mean 0 and variance 1 when ν is large.
Whether these approximations are sufficiently accurate depends on the purpose for which they are needed, and the rate of convergence to the normal distribution. It is typically the case that such approximations are less accurate in the tails of the distribution.
A general upper bound for the approximation error in the central limit theorem is given by the Berry–Esseen theorem, improvements of the approximation are given by the Edgeworth expansions.
Operations on normal deviates
The family of normal distributions is closed under linear transformations: if X is normally distributed with mean μ and standard deviation σ, then the variable Y = aX + b, for any real numbers a and b, is also normally distributed, with mean aμ + b and standard deviation |a|σ.
Also if X_{1} and X_{2} are two independent normal random variables, with means μ_{1}, μ_{2} and standard deviations σ_{1}, σ_{2}, then their sum X_{1} + X_{2} will also be normally distributed,^{[proof]} with mean μ_{1} + μ_{2} and variance .
In particular, if X and Y are independent normal deviates with zero mean and variance σ^{2}, then X + Y and X − Y are also independent and normally distributed, with zero mean and variance 2σ^{2}. This is a special case of the polarization identity.^{[22]}
Also, if X_{1}, X_{2} are two independent normal deviates with mean μ and deviation σ, and a, b are arbitrary real numbers, then the variable
is also normally distributed with mean μ and deviation σ. It follows that the normal distribution is stable (with exponent α = 2).
More generally, any linear combination of independent normal deviates is a normal deviate.
Infinite divisibility and Cramér's theorem
For any positive integer n, any normal distribution with mean μ and variance σ^{2} is the distribution of the sum of n independent normal deviates, each with mean μ/n and variance σ^{2}/n. This property is called infinite divisibility.^{[23]}
Conversely, if X_{1} and X_{2} are independent random variables and their sum X_{1} + X_{2} has a normal distribution, then both X_{1} and X_{2} must be normal deviates.^{[24]}
This result is known as Cramér's decomposition theorem, and is equivalent to saying that the convolution of two distributions is normal if and only if both are normal. Cramér's theorem implies that a linear combination of independent non-Gaussian variables will never have an exactly normal distribution, although it may approach it arbitrarily close.^{[25]}
Bernstein's theorem
Bernstein's theorem states that if X and Y are independent and X + Y and X − Y are also independent, then both X and Y must necessarily have normal distributions.^{[26]}^{[27]}
More generally, if X_{1}, …, X_{n} are independent random variables, then two distinct linear combinations ∑a_{k}X_{k} and ∑b_{k}X_{k} will be independent if and only if all X_{k}'s are normal and ∑a_{k}b_{k}Template:SubSup = 0, where Template:SubSup denotes the variance of X_{k}.^{[26]}
Other properties
Related distributions
Operations on a single random variable
If X is distributed normally with mean μ and variance σ^{2}, then
- The exponential of X is distributed log-normally: e^{X} ~ ln(N (μ, σ^{2})).
- The absolute value of X has folded normal distribution: |X| ~ N_{f} (μ, σ^{2}). If μ = 0 this is known as the half-normal distribution.
- The square of X/σ has the noncentral chi-squared distribution with one degree of freedom: X^{2}/σ^{2} ~ χ^{2}_{1}(μ^{2}/σ^{2}). If μ = 0, the distribution is called simply chi-squared.
- The distribution of the variable X restricted to an interval [a, b] is called the truncated normal distribution.
- (X − μ)^{−2} has a Lévy distribution with location 0 and scale σ^{−2}.
Combination of two independent random variables
If X_{1} and X_{2} are two independent standard normal random variables with mean 0 and variance 1, then
- Their sum and difference is distributed normally with mean zero and variance two: X_{1} ± X_{2} ∼ N(0, 2).
- Their product Z = X_{1}·X_{2} follows the "product-normal" distribution^{[28]} with density function f_{Z}(z) = π^{−1}K_{0}(|z|), where K_{0} is the modified Bessel function of the second kind. This distribution is symmetric around zero, unbounded at z = 0, and has the characteristic function φ_{Z}(t) = (1 + t^{ 2})^{−1/2}.
- Their ratio follows the standard Cauchy distribution: X_{1} ÷ X_{2} ∼ Cauchy(0, 1).
- Their Euclidean norm has the Rayleigh distribution.
Combination of two or more independent random variables
- If X_{1}, X_{2}, …, X_{n} are independent standard normal random variables, then the sum of their squares has the chi-squared distribution with n degrees of freedom
- If X_{1}, X_{2}, …, X_{n} are independent normally distributed random variables with means μ and variances σ^{2}, then their sample mean is independent from the sample standard deviation,^{[29]} which can be demonstrated using Basu's theorem or Cochran's theorem.^{[30]} The ratio of these two quantities will have the Student's t-distribution with n − 1 degrees of freedom:
- If X_{1}, …, X_{n}, Y_{1}, …, Y_{m} are independent standard normal random variables, then the ratio of their normalized sums of squares will have the F-distribution with (n, m) degrees of freedom:^{[31]}
Operations on the density function
The split normal distribution is most directly defined in terms of joining scaled sections of the density functions of different normal distributions and rescaling the density to integrate to one. The truncated normal distribution results from rescaling a section of a single density function.
Extensions
The notion of normal distribution, being one of the most important distributions in probability theory, has been extended far beyond the standard framework of the univariate (that is one-dimensional) case (Case 1). All these extensions are also called normal or Gaussian laws, so a certain ambiguity in names exists.
- The multivariate normal distribution describes the Gaussian law in the k-dimensional Euclidean space. A vector X ∈ R^{k} is multivariate-normally distributed if any linear combination of its components ∑Template:Sua_{j} X_{j} has a (univariate) normal distribution. The variance of X is a k×k symmetric positive-definite matrix V. The multivariate normal distribution is a special case of the elliptical distributions. As such, its iso-density loci in the k = 2 case are ellipses and in the case of arbitrary k are ellipsoids.
- Rectified Gaussian distribution a rectified version of normal distribution with all the negative elements reset to 0
- Complex normal distribution deals with the complex normal vectors. A complex vector X ∈ C^{k} is said to be normal if both its real and imaginary components jointly possess a 2k-dimensional multivariate normal distribution. The variance-covariance structure of X is described by two matrices: the variance matrix Γ, and the relation matrix C.
- Matrix normal distribution describes the case of normally distributed matrices.
- Gaussian processes are the normally distributed stochastic processes. These can be viewed as elements of some infinite-dimensional Hilbert space H, and thus are the analogues of multivariate normal vectors for the case k = ∞. A random element h ∈ H is said to be normal if for any constant a ∈ H the scalar product (a, h) has a (univariate) normal distribution. The variance structure of such Gaussian random element can be described in terms of the linear covariance operator K: H → H. Several Gaussian processes became popular enough to have their own names:
- Gaussian q-distribution is an abstract mathematical construction that represents a "q-analogue" of the normal distribution.
- the q-Gaussian is an analogue of the Gaussian distribution, in the sense that it maximises the Tsallis entropy, and is one type of Tsallis distribution. Note that this distribution is different from the Gaussian q-distribution above.
One of the main practical uses of the Gaussian law is to model the empirical distributions of many different random variables encountered in practice. In such case a possible extension would be a richer family of distributions, having more than two parameters and therefore being able to fit the empirical distribution more accurately. The examples of such extensions are:
- Pearson distribution— a four-parametric family of probability distributions that extend the normal law to include different skewness and kurtosis values.
Normality tests
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Normality tests assess the likelihood that the given data set {x_{1}, …, x_{n}} comes from a normal distribution. Typically the null hypothesis H_{0} is that the observations are distributed normally with unspecified mean μ and variance σ^{2}, versus the alternative H_{a} that the distribution is arbitrary. Many tests (over 40) have been devised for this problem, the more prominent of them are outlined below:
- "Visual" tests are more intuitively appealing but subjective at the same time, as they rely on informal human judgement to accept or reject the null hypothesis.
- Q-Q plot— is a plot of the sorted values from the data set against the expected values of the corresponding quantiles from the standard normal distribution. That is, it's a plot of point of the form (Φ^{−1}(p_{k}), x_{(k)}), where plotting points p_{k} are equal to p_{k} = (k − α)/(n + 1 − 2α) and α is an adjustment constant, which can be anything between 0 and 1. If the null hypothesis is true, the plotted points should approximately lie on a straight line.
- P-P plot— similar to the Q-Q plot, but used much less frequently. This method consists of plotting the points (Φ(z_{(k)}), p_{k}), where . For normally distributed data this plot should lie on a 45° line between (0, 0) and (1, 1).
- Shapiro-Wilk test employs the fact that the line in the Q-Q plot has the slope of σ. The test compares the least squares estimate of that slope with the value of the sample variance, and rejects the null hypothesis if these two quantities differ significantly.
- Normal probability plot (rankit plot)
- Moment tests:
- Empirical distribution function tests:
- Lilliefors test (an adaptation of the Kolmogorov–Smirnov test)
- Anderson–Darling test
Estimation of parameters
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It is often the case that we don't know the parameters of the normal distribution, but instead want to estimate them. That is, having a sample (x_{1}, …, x_{n}) from a normal N(μ, σ^{2}) population we would like to learn the approximate values of parameters μ and σ^{2}. The standard approach to this problem is the maximum likelihood method, which requires maximization of the log-likelihood function:
Taking derivatives with respect to μ and σ^{2} and solving the resulting system of first order conditions yields the maximum likelihood estimates:
Estimator is called the sample mean, since it is the arithmetic mean of all observations. The statistic is complete and sufficient for μ, and therefore by the Lehmann–Scheffé theorem, is the uniformly minimum variance unbiased (UMVU) estimator.^{[32]} In finite samples it is distributed normally:
The variance of this estimator is equal to the μμ-element of the inverse Fisher information matrix . This implies that the estimator is finite-sample efficient. Of practical importance is the fact that the standard error of is proportional to , that is, if one wishes to decrease the standard error by a factor of 10, one must increase the number of points in the sample by a factor of 100. This fact is widely used in determining sample sizes for opinion polls and the number of trials in Monte Carlo simulations.
From the standpoint of the asymptotic theory, is consistent, that is, it converges in probability to μ as n → ∞. The estimator is also asymptotically normal, which is a simple corollary of the fact that it is normal in finite samples:
The estimator is called the sample variance, since it is the variance of the sample (x_{1}, …, x_{n}). In practice, another estimator is often used instead of the . This other estimator is denoted s^{2}, and is also called the sample variance, which represents a certain ambiguity in terminology; its square root s is called the sample standard deviation. The estimator s^{2} differs from by having (n − 1) instead of n in the denominator (the so-called Bessel's correction):
The difference between s^{2} and becomes negligibly small for large n's. In finite samples however, the motivation behind the use of s^{2} is that it is an unbiased estimator of the underlying parameter σ^{2}, whereas is biased. Also, by the Lehmann–Scheffé theorem the estimator s^{2} is uniformly minimum variance unbiased (UMVU),^{[32]} which makes it the "best" estimator among all unbiased ones. However it can be shown that the biased estimator is "better" than the s^{2} in terms of the mean squared error (MSE) criterion. In finite samples both s^{2} and have scaled chi-squared distribution with (n − 1) degrees of freedom:
The first of these expressions shows that the variance of s^{2} is equal to 2σ^{4}/(n−1), which is slightly greater than the σσ-element of the inverse Fisher information matrix . Thus, s^{2} is not an efficient estimator for σ^{2}, and moreover, since s^{2} is UMVU, we can conclude that the finite-sample efficient estimator for σ^{2} does not exist.
Applying the asymptotic theory, both estimators s^{2} and are consistent, that is they converge in probability to σ^{2} as the sample size n → ∞. The two estimators are also both asymptotically normal:
In particular, both estimators are asymptotically efficient for σ^{2}.
By Cochran's theorem, for normal distributions the sample mean and the sample variance s^{2} are independent, which means there can be no gain in considering their joint distribution. There is also a reverse theorem: if in a sample the sample mean and sample variance are independent, then the sample must have come from the normal distribution. The independence between and s can be employed to construct the so-called t-statistic:
This quantity t has the Student's t-distribution with (n − 1) degrees of freedom, and it is an ancillary statistic (independent of the value of the parameters). Inverting the distribution of this t-statistics will allow us to construct the confidence interval for μ;^{[33]} similarly, inverting the χ^{2} distribution of the statistic s^{2} will give us the confidence interval for σ^{2}:^{[34]}
where t_{k,p} and Template:SubSup are the p^{th} quantiles of the t- and χ^{2}-distributions respectively. These confidence intervals are of the confidence level 1 − α, meaning that the true values μ and σ^{2} fall outside of these intervals with probability (or significance level) α. In practice people usually take α = 5%, resulting in the 95% confidence intervals. The approximate formulas in the display above were derived from the asymptotic distributions of and s^{2}. The approximate formulas become valid for large values of n, and are more convenient for the manual calculation since the standard normal quantiles z_{α/2} do not depend on n. In particular, the most popular value of α = 5%, results in |z_{0.025}| = 1.96.
Bayesian analysis of the normal distribution
Bayesian analysis of normally distributed data is complicated by the many different possibilities that may be considered:
- Either the mean, or the variance, or neither, may be considered a fixed quantity.
- When the variance is unknown, analysis may be done directly in terms of the variance, or in terms of the precision, the reciprocal of the variance. The reason for expressing the formulas in terms of precision is that the analysis of most cases is simplified.
- Both univariate and multivariate cases need to be considered.
- Either conjugate or improper prior distributions may be placed on the unknown variables.
- An additional set of cases occurs in Bayesian linear regression, where in the basic model the data is assumed to be normally distributed, and normal priors are placed on the regression coefficients. The resulting analysis is similar to the basic cases of independent identically distributed data, but more complex.
The formulas for the non-linear-regression cases are summarized in the conjugate prior article.
Sum of two quadratics
Scalar form
The following auxiliary formula is useful for simplifying the posterior update equations, which otherwise become fairly tedious.
This equation rewrites the sum of two quadratics in x by expanding the squares, grouping the terms in x, and completing the square. Note the following about the complex constant factors attached to some of the terms:
- The factor has the form of a weighted average of y and z.
- This shows that this factor can be thought of as resulting from a situation where the reciprocals of quantities a and b add directly, so to combine a and b themselves, it's necessary to reciprocate, add, and reciprocate the result again to get back into the original units. This is exactly the sort of operation performed by the harmonic mean, so it is not surprising that is one-half the harmonic mean of a and b.
Vector form
A similar formula can be written for the sum of two vector quadratics: If x, y, z are vectors of length k, and A and B are symmetric, invertible matrices of size , then
where
Note that the form x′ A x is called a quadratic form and is a scalar:
In other words, it sums up all possible combinations of products of pairs of elements from x, with a separate coefficient for each. In addition, since , only the sum matters for any off-diagonal elements of A, and there is no loss of generality in assuming that A is symmetric. Furthermore, if A is symmetric, then the form .
Sum of differences from the mean
Another useful formula is as follows:
With known variance
For a set of i.i.d. normally distributed data points X of size n where each individual point x follows with known variance σ^{2}, the conjugate prior distribution is also normally distributed.
This can be shown more easily by rewriting the variance as the precision, i.e. using τ = 1/σ^{2}. Then if and we proceed as follows.
First, the likelihood function is (using the formula above for the sum of differences from the mean):
Then, we proceed as follows: