exponentialCdf¶

Evaluates the exponential cumulative distribution function (CDF).

Synopsis¶

exponentialCdf (x, b)

Required Arguments¶

float x (Input): Argument for which the exponential CDF is to be evaluated. x must be non-negative.
float b (Input): Scale parameter of the exponential CDF. b must be positive.

Return Value¶

The probability that an exponential random variable takes a value less than or equal to x. A value of NaN is returned if an input value is in error.

Description¶

The function exponentialCdf evaluates the exponential cumulative distribution function (CDF). This function is a special case of the gamma CDF

\[G(x) = \frac{1}{\Gamma(a)} \int_0^x e^{-\tfrac{t}{b}} t^{a-1} dt\]

Setting \(a=1\) and applying the scale parameter b = b yields the exponential CDF

\[F(x) = \int_0^x e^{-\tfrac{t}{b}} dt = 1 - e^{-\tfrac{x}{b}}\]

This relationship between the gamma and exponential CDFs is used by exponentialCdf.

Example¶

In this example, we evaluate the exponential CDF at x = 2.0, b = 1.0.

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

x = 2.0
b = 1.0

p = exponentialCdf(x, b)
print("The probability that exponential random ")
print("variable X with scale parameter b = %3.1f" % b)
print("is less than or equal to %3.1f" % x)
print("is %6.4f" % p)

Output¶

The probability that exponential random 
variable X with scale parameter b = 1.0
is less than or equal to 2.0
is 0.8647