EX 3.38 traps

• Definition:
  • Let $X$ denote number of traps (defects of a certain kind)
• Distribution:
  • Follows a Poisson distribution
    • Parameter $\mu =2$
• The probability that there are exactly three traps is $P\left(X=3\right)=p\left(3;2\right)=\frac{e^{-2}{2}^3}{3!}=.180,$
• the probability that there are at most three traps is

$$\begin{array}{rl}P(X\le 3)& =F(3;2)\\ & =\sum _{x=0}^3\frac{e^{-2}{2}^x}{x!}\\ & =0.135+0.271+0.271+0.180\\ & =0.857\end{array}$$

• This latter cumulative probability is found at the intersection of the $\mu =2$ column and the $x=3$ row of Appendix Table A.2,
• whereas $p\left(3;2\right)=F\left(3;2\right)-F\left(2;2\right)=.857-.677=.180$
• the difference between two consecutive entries in the $\mu =2$ column of the cumulative Poisson table.