Note

Note that there are five types of callouts, including: note, warning, important, tip, and caution.

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Poisson Distribution

• SEE Taylor, binomial to Poisson
• BMLS
• PSU 414
• wikipedia
• http://www.econometricsbysimulation.com/search?q=poisson

Let $\chi$ be rv, counting of events , 0, 1, 2 … per interval. $\chi$ might be

• counts per unit time, distance.
• typos per page.
• cars passing per unit time.
• ATM customers per hour.

Motivating Poisson:

Suppose, on average, receive 9 letters in mail each day. And suppose can model post office as poisson. This addresses question: how many letters will I received today? Then expect sd to be sqrt(9) and so expect actual number of letters to vary between 3 and 12 (2 sd).

$$\begin{align*} Pr(X = x) = {e^{-\lambda}\lambda^x}/{x!} \end{align*}$$

lambda = 9    # rate of 1 per unit
x  <- 0:20    # how many counts?
z  <- dpois(x = x, lambda = lambda)
w  <- dpois(x = x, lambda = 1)
plot(z, main ="Poisson Distribution: upto n=20 counts, compare lambda= c(1,9)",
     ylab = "dpois",
     xlab = "counts, 1:n",
     pch = 19,
     col = "plum" 
)
points(w, pch = 20, col="red")   # colors()

# plot(w, main ="Poisson Distribution: upto n=20 counts, lambda=1",
#      ylab = "dpois",
#      xlab = "counts, 1:n",
#      pch = 19,
#      col = 3,
#      col.axis = "5"