naginterfaces.library.stat.prob_​normal_​vector

naginterfaces.library.stat.prob_normal_vector(tail, x, xmu, xstd)

prob_normal_vector returns a number of one or two tail probabilities for the Normal distribution.

For full information please refer to the NAG Library document for g01sa

https://support.nag.com/numeric/nl/nagdoc_30.1/flhtml/g01/g01saf.html

Parameters

tailstr, length 1, array-like, shape $\left(\text{ltail}\right)$

Indicates which tail the returned probabilities should represent. Letting $Z$ denote a variate from a standard Normal distribution, and $z_i=\frac{x_i-{\mu }_i}{{\sigma }_i}$, then for $j=\left(mod(\text{i}-1,\text{ltail})\right)+1-1$, for $\text{i}=1,2,\dots ,max(\text{lx},\text{ltail},\text{lxmu},\text{lxstd})$:

$tail\left[j\right]=\text{'L'}$

The lower tail probability is returned, i.e., $p_i=P(Z\le z_i)$.

$tail\left[j\right]=\text{'U'}$

The upper tail probability is returned, i.e., $p_i=P(Z\ge z_i)$.

$tail\left[j\right]=\text{'C'}$

The two tail (confidence interval) probability is returned, i.e., $p_i=P(Z\le \left|z_i\right|)-P(Z\le -\left|z_i\right|)$.

$tail\left[j\right]=\text{'S'}$

The two tail (significance level) probability is returned, i.e., $p_i=P(Z\ge \left|z_i\right|)+P(Z\le -\left|z_i\right|)$.

xfloat, array-like, shape $\left(\text{lx}\right)$

$x_i$, the Normal variate values.

xmufloat, array-like, shape $\left(\text{lxmu}\right)$

${\mu }_i$, the means.

xstdfloat, array-like, shape $\left(\text{lxstd}\right)$

${\sigma }_i$, the standard deviations.

Returns

pfloat, ndarray, shape $(max(\text{lx},\text{ltail}))$

$p_i$, the probabilities for the Normal distribution.

ivalidint, ndarray, shape $(max(\text{lx},\text{ltail}))$

$ivalid[i-1]$ indicates any errors with the input arguments, with

$ivalid[i-1]=0$

No error.

$ivalid[i-1]=1$

On entry, invalid value supplied in $tail$ when calculating $p_i$.

$ivalid[i-1]=2$

On entry, ${\sigma }_i\le 0.0$.

Raises

NagValueError

(errno $2$)

On entry, $\text{ltail}=⟨value⟩$.

Constraint: $\text{ltail}>0$.

(errno $3$)

On entry, $\text{lx}=⟨value⟩$.

Constraint: $\text{lx}>0$.

(errno $4$)

On entry, $\text{lxmu}=⟨value⟩$.

Constraint: $\text{lxmu}>0$.

(errno $5$)

On entry, $\text{lxstd}=⟨value⟩$.

Constraint: $\text{lxstd}>0$.

Warns

NagAlgorithmicWarning

(errno $1$)

On entry, at least one value of $tail$ or $xstd$ was invalid.

Check $ivalid$ for more information.

Notes

The lower tail probability for the Normal distribution, $P(X_i\le x_i)$ is defined by:

$$P(X_i\le x_i)={\int }_{-\infty }^{x_i}{Z}_i\left(X_i\right)d{X}_i\text{,}$$

where

$$Z_i\left(X_i\right)=\frac{1}{\sqrt{2\pi {\sigma }_i^2}}{e}^{-{(X_i-{\mu }_i)}^2/\left(2{\sigma }_i^2\right)},-\infty <X_i<\infty \text{.}$$

The relationship

$$P(X_i\le x_i)=\frac{1}{2}erfc\left(\frac{-(x_i-{\mu }_i)}{\sqrt{2}{\sigma }_i}\right)$$

is used, where erfc is the complementary error function, and is computed using specfun.erfc_real.

When the two tail confidence probability is required the relationship

$$P(X_i\le \left|x_i\right|)-P(X_i\le -\left|x_i\right|)=erf\left(\frac{|x_i-{\mu }_i|}{\sqrt{2}{\sigma }_i}\right)\text{,}$$

is used, where erf is the error function, and is computed using specfun.erf_real.

The input arrays to this function are designed to allow maximum flexibility in the supply of vector arguments by re-using elements of any arrays that are shorter than the total number of evaluations required. See the G01 Introduction for further information.

References

NIST Digital Library of Mathematical Functions

Hastings, N A J and Peacock, J B, 1975, Statistical Distributions, Butterworth