Value of lambda: poisson process

Question

What is the meaning of the lambda in a Poisson process? How should I find lambda when solving exercises?

Suppose we're given that the time between two phone calls is exponentially distributed and is ten minutes on average.

Is $\lambda = 10 \, \text{ or } \, \frac{1}{10}$?

You can accept my answer if it helped you. – Dec 23 '17 at 14:54 — , Dec 23 '17 at 14:54

score 1 · Accepted Answer · answered Dec 23 '17 at 13:03

$\lambda$ is a parameter that captures the average number of events in an interval. For example, take for instance that the mailman observes that a post office, on average, receives $40$ letters per day. Here, what is $\lambda$? Because the average event rate is $40$ per day, $\lambda = 40$.

For more, you may want to check out: this, this and this.

score 1 · Answer 2 · answered Dec 23 '17 at 18:27

The parameter $\lambda$ in a Poisson process is commonly called the rate. So

the number of occurrences in a unit of time has a Poisson distribution with expected value $\lambda$, and
the gap between occurrences has an exponential distribution with expected value $\frac{1}{\lambda}$

Your example "we're given that the time between two phone calls is exponentially distributed and is ten minutes on average" is telling you $\frac{1}{\lambda} = 10$ minutes or $\frac16$ hours, and you need to take the reciprocal to get $\lambda$ as a rate

Value of lambda: poisson process

2 Answers2