geometricInverseCdf¶

Evaluates the inverse of the discrete geometric cumulative distribution function (CDF).

Synopsis¶

geometricInverseCdf (p, pin)

Required Arguments¶

float p (Input): Probability for which the inverse of the discrete geometric CDF is to be evaluated. p must be in the open interval (0, 1).
float pin (Input): Probability parameter of the discrete geometric CDF (the probability of success for each independent trial). pin must be in the open interval (0, 1).

Return Value¶

The probability that a discrete geometric random variable takes a value less than or equal to the returned value is the input probability, p. A value of -1 is returned if an input value is in error.

Description¶

The function geometricInverseCdf evaluates the inverse CDF of a discrete geometric random variable with parameter pin. The discrete geometric CDF is defined:

\[p = F(I|P) = \sum_{i=0}^{I} P(1-P)^i = 1-(1-P)^{I+1}\]

where the return value \(p=F(I|P)\) is the probability that up to I trials would be observed before observing a success, and input parameter P = pin is the probability of success for each independent trial. The discrete geometric inverse CDF is defined:

\[I = F^{-1}(p|P) = \left\lceil \frac{\log(1-p)}{\log(1-P)} - 1 \right\rceil\]

which is the smallest integer I such that the discrete geometric CDF is greater than or equal to input argument p = p, where 0 < p < 1, and input parameter P = pin.

Example¶

In this example, we evaluate the inverse probability function at pin = 0.25, p = 0.6835.

from __future__ import print_function
from numpy import *
from pyimsl.stat.geometricInverseCdf import geometricInverseCdf

pin = 0.25
p = 0.6835

ix = geometricInverseCdf(p, pin)

print("The probability that a discrete geometric random variable")
print("with probability parameter pin = %4.2f is less than or equal" % pin)
print("to %2i is %6.4f" % (ix, p))

Output¶

The probability that a discrete geometric random variable
with probability parameter pin = 0.25 is less than or equal
to  3 is 0.6835