Question

Emails arrive in an inbox according to a Poisson process with rate 20 emailsper hour. LetTbe the time at which the 3rd email arrives, measured in hours after acertain fixed starting time. FindP(T >0.1) without using calculus.Hint: Apply the count-time duality.

Answer 1

Answer:

$P(T>0.1h)=5(e^{-2})$

Step-by-step explanation:

We are going to solve this exercise using count-time duality.

The event $T>0.1h$ means that the time when third mail arrives is later than 0.1 h

This fact means that in the previously interval there have come two or fewer emails.

Let be X the random variable : ''The number of mails that arrived during the first 0.1 h''

Using count-time duality ⇒ $P(T>0.1h)=P(X\leq 2)$

X ~ Po (20. 0.1)

X ~ Po (2)

Now we calculate $P(X\leq 2)$

The probability function for X is :

$P(X=x)=\frac{m^{x}e^{-m}}{x!}$

Where m = λ

$P(X\leq 2)=P(X=0)+P(X=1)+P(X=2)$

$P(X\leq 2)=\frac{2^{0}e^{-2}}{0!}+\frac{2^{1}e^{-2}}{1!}+\frac{2^{2}e^{-2}}{2!}$

$P(X\leq 2)=e^{-2}+2e^{-2}+2e^{-2}=5(e^{-2})$

Finally, $P(T>0.1h)=5e^{-2}$