# What is the difference between a Poisson and an Exponential distribution?

For a Poisson distribution:

$$\mathsf{P}(X=x)=\frac{e^{-\mu}\times \mu^x}{x!}$$ where $\mu$ is the mean number of occurrences.

For an Exponential distribution:

$$f(x;\lambda) = \begin{cases} \lambda e^{-\lambda x} & x \ge 0 \\ 0 & x < 0 \end{cases}$$

where $\lambda$ is the rate parameter.

Apart from the fact that the formulas are obviously different, in layman's terms what is the difference between an Exponential and Poisson distribution? Or put in another way, why do we need them both? What does one of them do that the other doesn't? What is the difference between an Exponential distribution and an Exponential Density function?

And yes, this is all very new to me as I was never taught distribution theory.

Thanks.

• Time between two consecutive Poisson arrivals is distributed exponentially.
– A.S.
Commented Nov 19, 2015 at 7:28
• This doesn't really fit in (distribution-theory). Commented Nov 19, 2015 at 7:44
• @Brian Removed the tag, thanks for your advice Commented Nov 19, 2015 at 7:58

As A.S.'s comment indicates, both distributions relate to the same kind of process (a Poisson process), but they govern different aspects: The Poisson distribution governs how many events happen in a given period of time, and the exponential distribution governs how much time elapses between consecutive events.

By way of analogy, suppose that we have a different process, in which events occur exactly every $10$ seconds. Then the number of events that happen in a minute (i.e., $60$ seconds) is deterministically $6$, and the amount of time that elapses between consecutive events is, of course, deterministically $10$ seconds.

In contrast, in a Poisson process with a mean rate of one event every $10$ seconds (i.e., $\lambda = 1/10$), the number of events that happen in a minute is not deterministically $6$, but it has a mean of $6$. The exact distribution is given by the Poisson distribution:

$$P_k(t) = \frac{(\lambda t)^k}{k!} e^{-\lambda t}$$

where $t = 60$ seconds is the time window. Thus, the probability that no events occur in a minute is given by

$$P_0(t) = \frac{(6)^0}{0!} e^{-6} = e^{-6} \doteq 0.0024788$$

whereas the probability that $6$ events occur in a minute is given by

$$P_6(t) = \frac{(6)^6}{6!} e^{-6} = \frac{46656}{720} e^{-6} \doteq 0.16062$$

That is obviously much more likely, as you would expect.

Similarly, the time between events is also not deterministically $10$ seconds, but it has a mean of $10$ seconds. The actual time distribution is the exponential distribution, which can be specified using its CDF (cumulative distribution function)

$$F_T(t) \equiv P(T < t) = 1-e^{-\lambda t}$$

The CDF provides essentially the same information as the PDF (probability density function), whose formulation you gave in your question; in fact, the derivative of the CDF is the PDF. However, the CDF is sometimes easier to understand intuitively, so I'll explain using the CDF here.

In this case, the probability that the time between events is less than $10$ seconds is $F_T(10) = 1-e^{-1} \doteq 0.63212$, whereas the probability that the time between events is less than $60$ seconds is $F_T(60) = 1-e^{-6} \doteq 0.99752$. The probability that it is greater than $60$ seconds is $1-F_T(60) = e^{-6} \doteq 0.0024788$, and you'll notice this is equal to the probability that no events occur in a given minute. This is no coincidence; in a Poisson process, which is memoryless, the probability that the time between events is greater than a minute is naturally equal to the probability that no events occur in that minute!

• Thank you very much for your excellent answer, Can you make a distinction between an Exponential distribution and an Exponential Density function? I'm getting lost in the terminology, even simpler; I don't know what the difference between a distribution and a density function is. Commented Nov 19, 2015 at 8:03
• See my comment to Zelos Malum's answer. Commented Nov 19, 2015 at 8:03
• @BrianTung Nice answer (+1)! Does $\doteq$ mean the same as $\approx$? Commented Nov 19, 2015 at 8:25
• Yes, it does: "approximately equal to". Commented Nov 19, 2015 at 19:10

Exponential is a continuous distribution while poisson is a discrete one.

• Thanks for your answer; What is the difference between an Exponential distribution and an Exponential Density function? Commented Nov 19, 2015 at 7:59
• Exponential distribution just tells us the over all shape of distibution at every instance while the density function is the function that gives us the probability, usually it is from $-\infty$ to $x$ for $f(x)$ Commented Nov 19, 2015 at 8:02
• I'd say that an exponential density function (an example of a PDF) is one way to characterize an exponential distribution. The CDF in my answer is another way. Commented Nov 19, 2015 at 8:03
• @ZelosMalum: Haven't you got that confused with the CDF, which is indeed the probability that the variable holds any value between $-\infty$ and some dummy value $x$? Commented Nov 19, 2015 at 8:04