9.4.15  The exponential distribution

The probability density function for the exponential distribution: exponential exponentiald

The exponential distribution depends on one parameters, λ>0; the value of the density function at x ≥ 0 is exponential(λ,x) = λ e^{−λ x}. The exponential command computes the exponential distribution.
exponentiald is a synonym for exponential.

• exponential takes two arguments:
  • λ, a positive number (the parameter).
  • x, a positive number.
• exponential(λ,x) returns the value of the exponential density function with parameter λ at x; namely, exponential(λ,x) = λ e^{−λ x}.

Example.

Output:

The cumulative distribution function for the exponential distribution: exponential_cdf exponentiald_cdf

The exponential_cdf (or exponentiald_cdf) command computes the cumulative distribution function for the exponential distribution.

• exponential_cdf (or exponentiald_cdf) takes two arguments:
  • λ, a positive number (the parameter).
  • x, a positive number.
• exponential_cdf(λ,x) returns Prob(X ≤ x) for the exponential distribution with parameter λ.
• exponential_cdf(λ,x,y) returns Prob(x ≤ X ≤ y).

Examples.

• Input:
  exponential_cdf(2.3,3.2)
  Output:

  0.99936380154

• Input:
  exponential_cdf(2.3,0.9,3.2)
  Output:

  0.125549583246

The inverse distribution function for the exponential distribution: exponential_icdf exponentiald_icdf

The exponential_icdf (or exponentiald_icdf) command computes the inverse distribution for the exponential distribution.

• exponential_icdf (or exponentiald_icdf) takes two arguments:
  • λ, a positive number (the parameter).
  • h, a positive real number.
• exponential_icdf(λ,h) returns the inverse distribution for the exponential distribution with parameter λ; namely, the value of x for which Prob(X ≤ x) = h.

Example.
Input:

Output: