Poisson counting process

Question

Let $N_t$ be number of customers that arrived at shop until moment $t$. Let's say that shop opens at 9:00. $N_t$, $t\geq0$ is Poisson process with $\lambda=1$ per hour. What is the probability that, there will be at least 2 arrivals between 10:00 and 10:30?

We have time interval (10:00,10:30] (can it be interpreted as $(0,30]=(0,0.5]$ ?). Thus needed probability is:

$$ P(N(1/2)-N(0)\geq 2) = P(N(1/2)\geq 2) $$ $$ = 1-\exp(-0.5\cdot0)\frac{(0.5\cdot0)^0}{0!}-\exp(-0.5\cdot1)\frac{(0.5\cdot1)^1}{1!}=0.09 $$

Am I correct?

André Nicolas · Accepted Answer · 2013-06-12T14:34:02.607

0

The number of arrivals in a specified $1/2$ hour interval has Poisson distribution with parameter $0.5$. The probability that the number of arrivals is $\ge 2$ is, by reasoning similar to yours, equal to $$1-\left(e^{-0.5}\frac{(0.5)^0}{0!}+e^{-0.5}\frac{(0.5)^1}{1!}\right).$$ For some reason, you incorrectly had $(-0.5)^0$, which by accident is OK, and $(-0.5)^1$, which is not.

You also had an error in the part dealing with the probability that the number is $0$. You had $e^{(-0.5)(0)}$ as part of that expression, and it should be simply $e^{-0.5}$.

Remark: The probability that the number of arrivals in a $t$ hour interval is exactly $k$, in the general case with parameter $\lambda$, is $$e^{-\lambda t}\frac{(\lambda t)^k}{k!}.$$ In our case, $\lambda=1$ and $t=0.5$. You appear to be working with a different (and incorrect) formula.

edited Jun 12 '13 at 14:34

answered Jun 12 '13 at 08:54

André Nicolas

507,029

I edited errors and now it should be fine. I was in hurry when I made them and didn't have time to revise everything. Thanks for everything! – atomoutside Jun 12 '13 at 10:41
@atomoutside: Please check your second term, the term after the $1$. Makes no difference, but it should be $e^{-0.5}$, or equivalently $e^{-0.5}\frac{(0.5)^0}{0!}$. – André Nicolas Jun 12 '13 at 14:43
You mean $e^{-\lambda t}$ right? Thanks. – atomoutside Jun 13 '13 at 07:27
Yes, with $\lambda=1$ and $t=0.5$, so it is the same thing. – André Nicolas Jun 13 '13 at 07:32

Poisson counting process

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