Page 199 Exercise 13

The poisson distribution with parameter \(\lambda = .3\) has been assigned for the outcome of an experiment. Let \(X\) be the outcome function. Find \(P(X=0)\), \(P(X=1)\), and \(P(X>1)\)

Solution by Hand

Poisson Formula:

\[ P(X=x) = \frac{\lambda^x e^{-\lambda}}{x!} \]

\(P(X=0)\)

\[ P(X=0) = \frac{0.3^0 e^{-0.3}}{0!} = \frac{e^{-0.3}}{1} = e^{-0.3} \approx 0.741 \]

\(P(X=1)\)

\[ P(X=1) = \frac{0.3^1 e^{-0.3}}{1!} = 0.3e^{-0.3} \approx 0.222 \]

\(P(X>1)\)

\[ P(X>1) = 1 - P(x\geq 1) = 1 - P(X=0) - P(X=1) =\\ 1 - e^{-0.3} - 0.3e^{-0.3} = 1 - 1.3e^{-0.3} \approx 0.0369 \]

Solution with R

x1 = dpois(0, 0.3)
x2 = dpois(1, 0.3)
x3 = 1 - x1 - x2

x1
## [1] 0.7408182
x2
## [1] 0.2222455
x3
## [1] 0.03693631