# The exponential distribution

### 8.4.15 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.

exponential(2.1,3.5)

Output:

0.00134944395675

#### 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:

exponential_icdf(2.3,0.87)

Output:

0.887052534142