The exponential distribution is used to model Poisson processes, which are situations in which an object initially in state A can change to state B with constant probability per unit time λ. The time at which the state actually changes is described by an exponential random variable with parameter λ. Therefore, the integral from 0 to T over f is the probability that the object is in state B at time T.

The exponential distribution may be viewed as a continuous counterpart of the geometric distribution, which describes the number of Bernoulli trials necessary for a discrete process to change state. In contrast, the exponential distribution describes the time for a continuous process to change state.

Examples of variables that are approximately exponentially distributed are:

the time until you have your next car accident;
the time until you get your next phone call (assuming you get called many times a day, or get called by people from many different time zones);
the distance between mutations on a DNA strand;
the distance between roadkill;
the time until a radioactive particle decays;
the number of dice rolls until you roll 6 11 times in a row.

3 Properties

3.1 Quartiles

The quartiles of an Exponential(λ) random variable are as follows:

first quartile: ln(4/3)/λ
median: ln(2)/λ
third quartile: ln(4)/λ

3.2 Expectations

An Exponential(λ) random variable has the following properties:

mean: μ = 1/λ
varianceThis article is about mathematics. Alternate meaning: variance (land use). In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically: σ² = 1/λ²
skewnessIn probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. Roughly speaking, a distribution has positive skew if the positive tail is longer and negative skew if the negat: γ₁ = 2
kurtosis excess: γ₁ = 6
entropyBernoulli trial as a function of success probability. Entropy is a concept in thermodynamics (see thermodynamic entropy), statistical mechanics and information theory. The concepts of information and entropy have deep links with one another, although it t: H = 1 − ln(λ) nat s

3.3 Memorylessness

An important property of the exponential distribution is that it is memorylessIn probability theory, memorylessness is a property of certain probability distributions: the exponential distributions and the geometric distributions. Discrete memorylessness Suppose X is a discrete random variable whose values lie in the set { 0, 1, 2,. This means that if a random variable T is exponentially distributed, its conditional probabilityThis article defines some terms which characterize probability distributions of two or more variables. Conditional probability is the probability of some event A assuming event B''. Conditional probability is written P ''A ''B , and is read "the probabili obeys

This says that the conditional probabilityThis article defines some terms which characterize probability distributions of two or more variables. Conditional probability is the probability of some event A assuming event B''. Conditional probability is written P ''A ''B , and is read "the probabili that we need to wait, for example, more than another 10 seconds before the first arrival, given that the first arrival has not yet happened after 30 seconds, is no different from the initial probability that we need to wait more than 10 seconds for the first arrival. This is often misunderstood by students taking courses on probability: the fact that P(T > 40 | T > 30) = P(T > 10) does not mean that the events T > 40 and T > 10 are independent. To summarize: "memorylessness" of the probability distribution of the waiting time T until the first arrival means

It does not mean

(That would be independence. These two events are not independent.)

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