Logisdf

Purpose

Logistic distribution.

Synopsis

prob = logisdf(function,x,a,b)

Description

Estimates cumulative distribution function (cumulative, cdf), probability density function (density, pdf), quantile (inverse of cdf), or random numbers for a Logistic distribution.

This distribution is a common alternative to the normal distribution. It is symmetric and many times used when data represents midpoints of interval data (data collected in such a way that a range instead or an exact value is collected). The variance may be smaller, equal, or larger than the mean for this distribution.

$f(x)={\frac {\exp \left[-\left(x-a\right)/b\right]}{b\left\{1+\exp \left[-\left(x-a\right)/b\right]\right\}^{2}}}$

$F(x)={\frac {1}{2}}\left\{1+\tanh \left[{\frac {1}{2}}\left(x-a\right)/b\right]\right\}$

Inputs

• function = [ {'cumulative'} | 'density' | 'quantile' | 'random' ], defines the functionality to be used. Note that the function recognizes the first letter of each string so that the string could be: [ 'c' | 'd' | 'q' | 'r' ].

• x = matrix in which the sample data is stored, in the interval (-inf,inf).

for function=quantile - matrix with values in the interval (0,1).

for function=random - vector indicating the size of the random matrix to create.

• a = mean parameter (real).

• b = standard deviation parameter (real and positive).

Note: If inputs (x, a, and b) are not equal in size, the function will attempt to resize all inputs to the largest input using the RESIZE function.

Note: Functions will typically allow input values outside of the acceptable range to be passed but such values will return NaN in the results.

Examples

Cumulative

>> prob = logisdf('c',0.99,1,2)
prob =
    0.4988
>> x    = [0:0.1:10];
>> plot(x,logisdf('c',x,1,2),'b-',x,logisdf('c',x,3,.5),'r-')

Density

>> prob = logisdf('d',0.99,1,2)
prob =
    0.1250
>> x    = [0:0.1:10];
>> plot(x,logisdf('d',x,2,1),'b-',x,logisdf('d',x,0.5,1),'r-')

Quantile

>> prob = logisdf('q',0.99,1,2)
prob =
   10.1902

Random

>> prob = logisdf('r',[4 1],2,1)
ans =
    0.4549
    0.4638
    0.3426
    0.5011

See Also

betadf, cauchydf, chidf, expdf, gammadf, gumbeldf, laplacedf, lognormdf, normdf, paretodf, raydf, triangledf, unifdf, weibulldf