Other exponential probabilities

The cumulative distribution function, \(F(x)\), can be used to find the probability that an exponentially distributed variable, \(X\), lies within any two limits

\[ \begin{align} P(a \lt X \lt b) \;\;=\;\; P(X \lt b) - P(X \lt a) \;\;&=\;\; F(b) - F(a) \\ &=\;\; e^{-a\lambda} - e^{-b\lambda} \end{align} \]

Example

An organisation's web site is accessed at a constant rate of 20 times per hour between 9am and 5pm. After an update to the site at 10am, what is the probability that the first access is between 10:05am and 10:10am?

If the first access is \(X\) minutes after 10am,

\[ X \;\; \sim \; \; \ExponDistn(\lambda = \frac{20}{60} \text{ accesses per minute}) \]

The required probability is

\[ P(5 < X < 10) \;\;=\;\; F(10) - F(5) \;\;=\;\; e^{-5\times \frac{20}{60}} - e^{-10\times \frac{20}{60}} \;\;=\;\; 0.1532 \]

Conditional probabilities

The exponential distribution has an important property called its memoryless property.

Memoryless property of exponential distribution

If \(X \sim \ExponDistn(\lambda) \),

\[ P(X > s+t \;|\; X > s) \;\; = \; \; P(X >t) \quad\quad \text{for all }s,t \ge 0\]

In other words, knowing that there were no events in the interval \((0, s]\) gives no information about how long it will take for the first event to occur after time \(s\).

This is a direct consequence of the properties of a Poisson process — the occurrence of events in any time interval such as \((s, t]\) is independent of the occurrence of events in other intervals that do not overlap, such as \((0, s]\).

Alternatively the result can be proved mathematically.

\[ \begin{align} P(X > s+t \;|\; X > s) \;\;&=\;\; \frac {P(X > s+t \textbf{ and }X > s)} {P(X > s)} \\ &=\;\; \frac {P(X > s+t)} {P(X > s)} \\ &=\;\; \frac {e^{-(s+t)\lambda}}{e^{-s\lambda}} \;\;=\;\; e^{-t\lambda} \end{align} \]

Time between events

The memoryless property of a Poisson process with rate \(\lambda\) also means that if it is known that an event happened at time \(s\), the time from then until the next event in the process also has an \(\ExponDistn(\lambda) \) distribution.

The exponential distribution therefore also describes the time between events in a Poisson process.