Answer:
[tex]a. \ P(X\leq 8)=0.932\\b. \ P(X=8)=0.065\\c. \ P(9\leq X)=0.068\\d. \ P(5\leq X \leq 8)=0.491\\[/tex]
e. P(5<X<8)=0.251
Step-by-step explanation:
Given that boiler follows a Poisson distribution,$\sim$
[tex]X $\sim$Poi(\mu=5)[/tex]
Poisson Distribution formula is given by the expression:-
[tex]p(x,\mu)=\frac{e^{-\mu}\mu^x}{x!}[/tex]
Our probabilities will be calculated as below:
a.
[tex]P(X=x)=\frac{5^xe^{-5}}{x!}\\P(X\leq 8)=P(X=0)+P(X=1)+...+P(X=8)\\=\frac{5^0e^{-5}}{1!}+...+\frac{5^8e^{-5}}{8!}\\=0.006738+...+0.065278\\=0.961[/tex]
b.
[tex]P(X=8)=\frac{5^8e^{-5}}{8!}\\=0.065[/tex]
c.
[tex]P(9\leq X)=1-P(X=0)-P(X=1)-...-P(X=8)\\=0.068[/tex]
d.
[tex]P(5\leq X\leq 8)=P(X=5)+P(X=6)+P(X=7)+P(X=8)\\=0.491[/tex]
e.P(5<X<8)
[tex]P(X=6)+P(X=7)\\=0.251[/tex]