is used, where erfc is the complementary error function, and is computed using nag_specfun_erfc_real (s15ad). For the upper tail probability the relationship

P (X \geq x) = P (X \leq - x)

is used and for the two tail significance level probability twice the probability obtained from the absolute value of

x

is returned.

When the two tail confidence probability is required the relationship

P (X \leq |x|) - P (X \leq - |x|) = \erf (\frac{|x|}{\sqrt{2}}),

is used, where erf is the error function, and is computed using nag_specfun_erf_real (s15ae).

References

Abramowitz M and Stegun I A (1972) Handbook of Mathematical Functions (3rd Edition) Dover Publications

Hastings N A J and Peacock J B (1975) Statistical Distributions Butterworth

Parameters

Compulsory Input Parameters

1: $x$ – double scalar: $x$ , the value of the standard Normal variate.

Optional Input Parameters

1: $tail$ – string (length ≥ 1)

Default:

'L'

Indicates which tail the returned probability should represent.

$tail ='L'$: The lower tail probability is returned, i.e., $P (X \leq x)$ .
$tail ='U'$: The upper tail probability is returned, i.e., $P (X \geq x)$ .
$tail ='S'$: The two tail (significance level) probability is returned, i.e., $P (X \geq |x|) + P (X \leq - |x|)$ .
$tail ='C'$: The two tail (confidence interval) probability is returned, i.e., $P (X \leq |x|) - P (X \leq - |x|)$ .

Constraint:

tail ='L'

'U'

'S'

'C'

Output Parameters

1: $result$ – double scalar: The result of the function.
2: $ifail$ – int64int32nag_int scalar: $ifail = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:

ifail \neq 0

, then nag_stat_prob_normal (g01ea) returns

0.0

$ifail = 1$

On entry,

tail \neq'L'

'U'

'S'

'C'

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.

$ifail = - 999$: Dynamic memory allocation failed.

Accuracy

Accuracy is limited by machine precision. For detailed error analysis see nag_specfun_erfc_real (s15ad) and nag_specfun_erf_real (s15ae).

Further Comments

None.

Example

Four values of tail and x are input and the probabilities calculated and printed.

Open in the MATLAB editor: g01ea_example

function g01ea_example


fprintf('g01ea example results\n\n');

% Probability for Normal distribution
x     = 1.96;
tail = {'Lower'; 'Upper'; 'Confidence'; 'Significance'};

fprintf('  Tail    x      probability\n');
for j = 1:size(tail,1);

  [p, ifail] = g01ea( ...
                      x,'tail',tail{j});

  fprintf('%4s%8.2f%12.4f\n',tail{j}(1),x,p);
end

g01ea example results

  Tail    x      probability
   L    1.96      0.9750
   U    1.96      0.0250
   C    1.96      0.9500
   S    1.96      0.0500

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox