Intuition behind the Poisson distribution parameter lambda

Question

Why do we define $\lambda = np$ when deriving the Poisson distribution from the Binomial distribution? How do you come up with this definition? I couldn't find a good intuitive explanation of why this is done. My textbook and professors just stated this definition and then made a derivation.

Answer 1

You are not 'deriving' the Poisson distribution from the binomial distribution. In some circumstances (especially when binomial $p$ is small and binomial $n$ is large) it is possible to find a Poisson distribution that approximates the a binomial distribution. Such an approximation works best if the Poisson mean $\lambda$ matches the binomial mean $np.$

Here is a table, made using R, comparing selected probabilities for $\mathsf{Binom}(n = 200, p = 0.1)$ and $\mathsf{Pois}(\lambda = 20).$

(You can ignore line numbers in [ ]s.)

k = 5:35
b.pdf = round(dbinom(k, 200, .1), 4)
p.pdf = round(dpois(k, 20), 4)
abs.dif = abs(b.pdf - p.pdf) 
cbind(k, b.pdf, p.pdf, abs.dif)

Table:

       k  b.pdf  p.pdf abs.dif
 [1,]  5 0.0000 0.0001  0.0001
 [2,]  6 0.0001 0.0002  0.0001
 [3,]  7 0.0003 0.0005  0.0002
 [4,]  8 0.0009 0.0013  0.0004
 [5,]  9 0.0021 0.0029  0.0008
 [6,] 10 0.0045 0.0058  0.0013
 [7,] 11 0.0087 0.0106  0.0019
 [8,] 12 0.0153 0.0176  0.0023
 [9,] 13 0.0245 0.0271  0.0026
[10,] 14 0.0364 0.0387  0.0023
[11,] 15 0.0501 0.0516  0.0015
[12,] 16 0.0644 0.0646  0.0002
[13,] 17 0.0775 0.0760  0.0015
[14,] 18 0.0875 0.0844  0.0031
[15,] 19 0.0931 0.0888  0.0043
[16,] 20 0.0936 0.0888  0.0048
[17,] 21 0.0892 0.0846  0.0046
[18,] 22 0.0806 0.0769  0.0037
[19,] 23 0.0693 0.0669  0.0024
[20,] 24 0.0568 0.0557  0.0011
[21,] 25 0.0444 0.0446  0.0002
[22,] 26 0.0332 0.0343  0.0011
[23,] 27 0.0238 0.0254  0.0016
[24,] 28 0.0163 0.0181  0.0018
[25,] 29 0.0108 0.0125  0.0017
[26,] 30 0.0068 0.0083  0.0015
[27,] 31 0.0042 0.0054  0.0012
[28,] 32 0.0024 0.0034  0.0010
[29,] 33 0.0014 0.0020  0.0006
[30,] 34 0.0008 0.0012  0.0004
[31,] 35 0.0004 0.0007  0.0003

In the following graphical comparison, differences in probabilities less than 0.003 are difficult to distinguish.

hdr="PDFs of BINOM(200, .1) [bars] and POIS(20)"
k  = 0:40
plot(k, dbinom(k,200,.1), type="h", lwd=2, 
      col="blue", main=hdr)
 points(k, dpois(k, 20), col="red")
 abline(h = 0, col="green2")
 abline(v = 0, col="green2")

Note: Binomial probabilities $X\sim\mathsf{Binom}(n,p)$ have $P(X > n) = 0.$ Technically, Poisson probabilities never reach $0,$ no matter how far into the right tail you go. But for practical purposes the probabilities become too small to be of importance beyond some finite point. In particular, for $Y \sim\mathsf{Pois}(\lambda= 20),$ we have $P(Y > 40) \approx 0.$ Markov's inequality does not give 'tight' bound, but does guarantee $P(Y \ge k\lambda) \le \lambda/k\lambda = 1/k.$

1 - ppois(40,20)
[1] 2.542632e-05