Normal distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ N(μ,σ^{2}) is used to indicate that the random variable X has the normal distribution with parameters μ and σ^{2}. A normal random variable X with mean μ and variance σ^{2} has probability density function

f(x) = \frac{1}{\sqrt{2π}σ} e^{− \begin{matrix}(x−μ) & ^{2} \\ 2σ & ^{2} \end{matrix}} − ∞ < x < ∞,

for −∞ < μ < ∞ and σ > 0. The normal distribution can be used for modeling adult heights, newborn baby weights, ball bearing diameters, etc. The normal distribution can be used to approximate the binomial distribution when n is large and p is close to 1/2. The normal distribution can also be used to approximate the Poisson distribution when n is large and p is small. The central limit theorem indicates that the normal distribution is useful for modeling random variables that can be thought of as a sum of several independent random variables. The probability density function for μ = 3 and two different values of σ is illustrated below.

PSfrag replacements

σ = 1
σ = 2
x
f(x)

The cumulative distribution function on the support of X is

F (x) = P(X ≤ x) = \frac{erf( \frac{x−μ}{\sqrt{2}σ}) + 1}{2} − ∞ < x < ∞,

where

erf(x) = \frac{2}{\sqrt{π}} ∫_{0}^{x}e^{−t^{2}}dt x > 0

and erf(−x) = −erf(x). The survivor function on the support of X is

S(x) = P(X ≥ x) = \frac{1 − erf( \frac{x−μ}{\sqrt{2}σ})}{2} − ∞ < x < ∞.

The hazard function on the support of X is

h(x) = \frac{f(x)}{S(x)} = −\frac{e^{− \frac{(x−μ)^{2}}{2σ^{2}}} \sqrt{2}}{σ\sqrt{π}(erf(\frac{x−μ}{\sqrt{2}σ}) − 1)} − ∞ < x < ∞.

The hazard function can be difficult to calculate for large values of x because the survivor function S(x) and the probability density function f(x) are small. Details and a fix in R are given at

http://stackoverflow.com/questions/39510213/calculating-hazard-function-in-r-for-the-standard-normal-distribution .

The cumulative hazard function on the support of X is mathematically intractable.

The inverse distribution function of X is

F^{−1} (u) = μ + σ\sqrt{2}erf^{−1} (2u − 1) 0 < u < 1.

The median of X is μ. The moment generating function of X is

M(t) = E [e^{tX}] = e^{t(tσ^{2}+2μ)/2} − ∞ < t < ∞.

The characteristic function of X is

ϕ(t) = E [e^{itX}] = e^{t(−tσ^{2}+2iμ)/2} − ∞ < t < ∞.

The population mean, variance, skewness, and kurtosis of X are

E[X] = μ  V [X] = σ^{2}  E [( \frac{X − μ}{σ} ) ^{3}] = 0  E [( \frac{X − μ}{σ} ) ^{4}] = 3.

APPL verification: The APPL statements

X := NormalRV(mu, sigma); CDF(X); SF(X); HF(X); IDF(X); Mean(X); Variance(X); Skewness(X); Kurtosis(X); MGF(X);

verify the cumulative distribution function, survivor function, hazard function, inverse distribution function, population mean, variance, skewness, kurtosis, and moment generating function.