Standard normal distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ N(0,1) is used to indicate that the random variable X has the standard normal distribution. A standard normal random variable X has probability density function

f(x) = \frac{e^{−x^{2}/2}}{\sqrt{2π}} − ∞ < x < ∞.

The standard normal random variable arises because a normal random variable with mean μ and variance σ^{2} can be standardized by subtracting μ, then dividing by σ. This means that only a single table is required for all calculations involving the normal distribution. The probability density function is illustrated below.

PSfrag replacements

x
f(x)

The cumulative distribution function is

F (x) = \frac{1}{2} + \frac{1}{2} erf( \frac{x}{\sqrt{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) = \frac{1}{2} − \frac{1}{2} erf( \frac{x}{\sqrt{2}}) − ∞ < x < ∞.

The hazard function on the support of X is

h(x) = −\begin{matrix}e^{− \frac{1}{2} x^{2}} & \sqrt{2} \\ \sqrt{π} & ( −1 + erf( \frac{x}{\sqrt{2}}) ) \end{matrix} − ∞ < x < ∞.

The cumulative hazard function on the support of X is

H(x) = −lnS(x) = ln(2)− ln( 1 − erf( \frac{x}{\sqrt{2}}) ) − ∞ < x < ∞.

The inverse distribution function of X is

F^{−1} (u) = \sqrt{2}(erf^{−1} (2u − 1)) 0 ≤ u ≤ 1.

The median and mode of X are 0.

The moment generating function of X is

M(t) = e^{t^{2}/2} − ∞ < t < ∞.

The characteristic function of X is

ϕ(t) = e^{−t^{2}/2} − ∞ < t < ∞.

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

E[X] = 0  V [X] = 1  E [( \frac{X − μ}{σ} ) ^{3}] = 0  E [( \frac{X − μ}{σ} ) ^{4}] = 3.

APPL verification: The APPL statements

X := StandardNormalRV(); 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.