Respuesta :
Answer:
[tex]E(X) = V(X) = 2.41[/tex]
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
e = 2.71828 is the Euler number
[tex]\mu[/tex] is the mean in the given interval.
In the Poisson distribution, the mean is the same as the variance.
P(X = 0) = 0.09
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
[tex]P(X = 0) = \frac{e^{-\mu}*\mu^{0}}{(0)!} = e^{-\mu}[/tex]
So
[tex]e^{-\mu} = 0.09[/tex]
We apply ln to both sides, which is the inverse operation of the exponential.
So
[tex]\ln{e^{-\mu}} = \ln{0.09}[/tex]
[tex]-\mu = -2.41[/tex]
Multiplying by -1
[tex]\mu = 2.41[/tex]
So
[tex]E(X) = V(X) = 2.41[/tex]
