Exponential distribution ( $Exp (λ)$ )

. The mixed joint density factors into those marginals, proving independence. The integer part is a geometric distribution on zero-based support, with expectation

1/ (e^{λ} - 1)

Laplace transform of an exponential distribution

 0  0

X \sim Exp (λ)

, then direct integration gives

E [e^{- s X}] = \frac{λ}{λ + s}, Re s > - λ .

(1)

Memorylessness of the exponential distribution

 0  0

T \sim Exp (μ)

, then

P (T > s + t ∣ T > s) = P (T > t) = e^{- μ t} .

(1)

Thus an exponential remaining lifetime has the same law regardless of the elapsed lifetime.

Uniform ratio of independent exponential variables

 0  0

A

and

B

are independent exponential variables with the same rate, then

A / (A + B)

is uniform on

(0, 1)

and is independent of

A + B

Competing exponential clocks

 0  0

For independent exponential variables

S_{i}

of rates

q_{i}

and

Q = \sum_{i} q_{i}

, let

T = min_{i} S_{i}

and let

K

be the minimizing index. Then

P (K = k, T \geq t) = \frac{q _{k}}{Q} e^{- Qt} .

(1)

Consequently

T

is exponential with rate

Q

P (K = k) = q_{k} / Q

, and

K

and

T

are independent.

Geometric sum of exponential variables

 0  0

N

is geometric on

{1, 2, \dots}

with parameter

p

, independently of unit-rate exponential variables

S_{i}

, then

\sum_{i = 1}^{N} S_{i}

(1)

is exponential with rate

p

. Its moment-generating function is

p / (p - θ)

for

θ < p

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The Exponential distribution is a continuous probability distribution often used to model the time between events in a Poisson process. This distribution is characterized by its memoryless property, which implies that the probability of an event occurring in the next instant is independent of how much time has already elapsed.

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