Respuesta :
Answer:
(a) The probability of the event (X > 84) is 0.007.
(b) The probability of the event (X < 64) is 0.483.
Step-by-step explanation:
The random variable X follows a Poisson distribution with parameter λ = 64.
The probability mass function of a Poisson distribution is:
[tex]P(X=x)=\frac{e^{-\lambda}\lambda^{x}}{x!};\ x=0, 1, 2, ...[/tex]
(a)
Compute the probability of the event (X > 84) as follows:
P (X > 84) = 1 - P (X ≤ 84)
[tex]=1-\sum _{x=0}^{x=84}\frac{e^{-64}(64)^{x}}{x!}\\=1-[e^{-64}\sum _{x=0}^{x=84}\frac{(64)^{x}}{x!}]\\=1-[e^{-64}[\frac{(64)^{0}}{0!}+\frac{(64)^{1}}{1!}+\frac{(64)^{2}}{2!}+...+\frac{(64)^{84}}{84!}]]\\=1-0.99308\\=0.00692\\\approx0.007[/tex]
Thus, the probability of the event (X > 84) is 0.007.
(b)
Compute the probability of the event (X < 64) as follows:
P (X < 64) = P (X = 0) + P (X = 1) + P (X = 2) + ... + P (X = 63)
[tex]=\sum _{x=0}^{x=63}\frac{e^{-64}(64)^{x}}{x!}\\=e^{-64}\sum _{x=0}^{x=63}\frac{(64)^{x}}{x!}\\=e^{-64}[\frac{(64)^{0}}{0!}+\frac{(64)^{1}}{1!}+\frac{(64)^{2}}{2!}+...+\frac{(64)^{63}}{63!}]\\=0.48338\\\approx0.483[/tex]
Thus, the probability of the event (X < 64) is 0.483.
