# Understanding $\lambda$ in the definition of Poisson distributions

I am trying to understand the meaning of $$\lambda$$ in Poisson distributions. I know that it is the average rate of occurrences of the event, but I have not been able to fully understand what that means.

In "A First Course in Probability" by Sheldon Ross, the author says that a Poisson distribution may be used as an approximation for a binomial distribution with parameters $$(n,p)$$ when $$n$$ is large and $$p$$ is small enough so that $$np$$ is of moderate size.

1. What does $$np$$ being of moderate size mean? What is considered as of moderate size?
2. Why does $$np$$ have to be of moderate size? Why can a Poisson distribution not be used as an approximation for a binomial distribution if $$np$$ is too big or too small?

Also, in other books, I read that a Poisson distribution is the limiting case of a binomial distribution when $$\lambda=np$$ is constant as $$n\to\infty$$.

Under what conditions, is $$\lambda$$ constant as $$n\to\infty$$?

I am new to probability. If someone can provide the intuition behind Poisson distributions (specifically about $$\lambda$$), I would greatly appreciate it.

• Personally, I have better understood what a Poisson distribution is when I saw Poisson processes as solutions to a differential equation as described in this answer. You need analysis techniques for understanding limit behaviors... Commented Apr 29, 2022 at 4:26
• Besides, you don't mention the exceedingly important connection of this discrete distribution with the "continuous" exponential distribution with parameter ... $\lambda$. See here Commented Apr 29, 2022 at 4:32
• Does this answer your question? What is the intuition behind the Poisson distribution's function? Commented Apr 29, 2022 at 6:07
• @KurtG. Your proposed duplicate does not answer the question here. OP is asking specifically, "Why does np have to be of moderate size?". [Thanks for comment, r.e.s.!] Commented May 5, 2022 at 18:20

This is a fantastic question, as it gets at the heart of a common misconception. The answer to the question "Why does $$np$$ have to be of moderate size?" is that it does not! The only thing that is needed for Poisson approximation is that $$p$$ is small.

To make this precise, we need a way to measure the quality of a probabilistic approximation. If $$X$$ and $$Y$$ are random variables, we define the total variation distance between them to be $$d_{\text{TV}}(X, Y)=\sup_{A} |P(X\in A)-P(Y\in A)|$$ If this distance is small, say it is at most $$\epsilon$$, then you can use the distribution of $$Y$$ to calculation the probability of an event involving $$X$$, and have an absolute error of at most $$\epsilon$$.

Theorem Let $$X\sim \text{Binomial}(n,p)$$, let $$\lambda=np$$, and let $$Y\sim \text{Poisson}(\lambda)$$. Then $$d_\text{TV}(X,Y)\le \min\{p,np^2\}\tag1$$

Therefore, as long as $$p$$ is small, the quality of the Poisson approximation is good.

This same idea is true in greater generality. For a Binomial distribution, there are $$n$$ independent and equiprobable events; we can drop the requirement of equiprobability. Let $$E_1,\dots,E_n$$ be independent events, with $$P(E_i)=p_i$$ for $$i\in \{1,\dots,n\}$$. If we let $$X$$ be the number of events which occur, and let $$Z$$ be a Poisson random variable with the same mean as $$X$$, meaning that $$Z\sim \text{Poisson}(\lambda)$$ with $$\lambda=\sum_{i=1}^n p_i$$, then

$$d_\text{TV}(X, Z)\le \min\left\{\frac1{\lambda}\sum_{i=1}^n p_i^2, \,\,\sum_{i=1}^n p_i^2\right\}\tag2$$ In the case where $$p_1=\dots=p_n=p$$, we recover $$(1)$$. For a proof of $$(2)$$, see Approximate Computations of Expectations, by Charles Stein. Equation $$(2)$$ appears on page 89.

• "Moderate" is not a precise term; it just means "neither larger nor small," but large and small are subjective.

• When people say that $$\lambda=np$$ is constant as $$n\to\infty$$, it just means that $$n$$ is getting larger, and for each particular instance of $$n$$, the corresponding value of $$p$$ is equal to $$\lambda/n$$. Saying

The Binomial$$(n,p)$$ distribution as $$n\to\infty$$ with $$\lambda=np$$ constant.

means the exact same thing as

The Binomial$$(n,\tfrac{\lambda}n)$$ distribution as $$n\to\infty$$.

So, Binomial$$(10,0.5)$$ is pretty close to a Poisson$$(5)$$ distribution, Binomial$$(50,0.1)$$ is even closer, Binomial$$(500,0.01)$$ is extremely close, etc.

• Thank you so much for your great answer. Can you also explain what it means to assume $\lambda$ is a constant as $n\to\infty$? In such cases where $n\to\infty$ implies $p\to0$, is $\lambda=np$ always a constant as $n\to\infty$?
– Koda
Commented May 6, 2022 at 2:53
• @kmiyazaki See edit. Commented May 6, 2022 at 14:41
• Thank you so much!
– Koda
Commented May 6, 2022 at 19:51