# Deriving the Poisson from the Binomial

In my notes I have the following explanation:

The probability function of the Poisson random variable is $$P_X(k)={\alpha}^k \frac{e^{-\alpha}}{k!}$$ A Poisson random variable with parameter $$\alpha$$.

X is the number of times that an event happens in a fixed period of time $$T$$(or space).

This distribution is related to the binomial distribution. And what's the difference between the parameters $$\alpha$$ and $$\lambda$$?

We can see this relation if we divide the interval $$T$$ in $$n$$ subintervals of length $${\Delta}t= \frac{T}{n}$$. For large $$n$$ we get a very little $${\Delta}t$$. Each event can only happens once in every subinterval $$n$$, with probability $$p={\lambda}{\Delta}t$$.

Now, the total number of events is a Binomial $$(n,p)$$ random variable that equals the original Poisson$$(\alpha)$$variable when n $$\to \infty$$, with $$\alpha= {\lambda}T$$

Why does the probability of every subinterval $$n$$ equal $$p={\lambda}{\Delta}t$$?

EDIT: And what's the difference between the parameters $$\alpha$$ and $$\lambda$$? Usually a Poisson distribution has a parameter $$\lambda$$. But here $$\alpha$$ is used instead, as the parameter of the distribution, and later, in the definiton of the probability of each subinterval $$n$$, ($$p={\lambda}T$$), $$\lambda$$ is used.

If I understand you correctly, it seems that according to your note, p is the rate of successes per time interval $$\Delta t$$ and n is the number of trials in that interval. If that is the case, then in each interval, you run $$n$$ trials and have a rate of success of $$\alpha$$, thus it should follow that the rate of success in each interval is $$p=\frac{\alpha}{n}$$. Substitute this back into your formula should give you an explanation
• Thank you for your answer. I think that $\alpha$ is the rate of successes per time interval $T$ instead of ${\Delta}t$ Then the substitution wouldn't work.Do you know what's the difference between the parameters $\alpha$ and $\lambda$? – roy212 Apr 26 at 7:26
• I'm sorry, I for mistyping. I just edited p is the rate of success of $\Delta$t. So if you run $n$ trials in the interval $T$ with the rate of success $\alpha$, the rate of success in one interval $\Delta$t is $\frac{\alpha}{n}$ – Joe Martin Apr 26 at 16:22