Stochastic Modelling for Systems Biology, Second Edition, 2nd Edition

86 PROBA BILITY MODELS

Also,

E



X

2



=

Z

∞

0

x

2

λe

−λx

dx =

2

λ

2

,

and so

Va r(X) =

2

λ

2

−

1

λ

2

=

1

λ

2

.

Note that this means the expectation and standard deviation are both 1/λ.

Notes

1. As λ incre ases, the proba bility of small values of X increases and the mean de-

creases.

2. The median m is given by

m =

log 2

λ

= log 2 E(X) < E(X) .

3. The expo nential distribution is often used to model times between random events

(such as biochemical reactions). Some of the reasons are given below.

Proposition 3.16 (memoryless property) If X ∼ Exp(λ), then for a ny s, t ≥ 0 we

have

P(X > (s + t)|X > t) = P(X > s) .

If we think of the exponential rand om quantity as representing the time to ...