Respuesta :
Answer:
a) 15.63% probability that 5 hits are received in a given minute.
b) 6.88% probability that 9 hits are received in 1.5 minutes.
c) 67.67% probability that fewer than 3 hits are received in a period of 30 seconds.
Step-by-step explanation:
In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
In which
x is the number of sucesses
[tex]e = 2.71828[/tex] is the Euler number
[tex]\mu[/tex] is the mean in the given time interval.
(a.) What is the probability that 5 hits are received in a given minute?
Mean rate of 4 per minute, which means that [tex]\mu = 4[/tex]
This is P(X = 5).
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
[tex]P(X = 5) = \frac{e^{-4}*(4)^{5}}{(5)!} = 0.1563[/tex]
15.63% probability that 5 hits are received in a given minute.
(b.) What is the probability that 9 hits are received in 1.5 minutes?
Mean rate of 4 per minute, so for 1.5 minutes, [tex]\mu = 4*1.5 = 6[/tex]
This is P(X = 9).
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
[tex]P(X = 9) = \frac{e^{-6}*(6)^{9}}{(9)!} = 0.0688[/tex]
6.88% probability that 9 hits are received in 1.5 minutes.
(c.) What is the probability that fewer than 3 hits are received in a period of 30 seconds?
Mean rate of 4 per minute, so for 30 seconds = 0.5 minutes, [tex]\mu = 4*0.5 = 2[/tex]
This is
[tex]P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)[/tex]
In which
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
[tex]P(X = 0) = \frac{e^{-2}*(2)^{0}}{(0)!} = 0.1353[/tex]
[tex]P(X = 1) = \frac{e^{-2}*(2)^{1}}{(1)!} = 0.2707[/tex]
[tex]P(X = 2) = \frac{e^{-2}*(2)^{2}}{(2)!} = 0.2707[/tex]
So
[tex]P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) = 0.1353 + 0.2707 + 0.2707 = 0.6767[/tex]
67.67% probability that fewer than 3 hits are received in a period of 30 seconds.
