Answer:
Both the mean and the variance are equal to 3.51
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.
The mean and the variance are equal, so [tex]\mu[/tex] is also the variance.
We have that:
[tex]P(X = 0) = 0.03[/tex]
[tex]P(X = 0) = \frac{e^{-\mu}*\mu^{0}}{(0)!} = e^{-\mu}[/tex]
So
[tex]e^{-\mu} = 0.03[/tex]
Applying ln to both sides
[tex]\ln{e^{-\mu}} = \ln{0.03}[/tex]
[tex]-\mu = -3.51[/tex]
Multiplying by -1
[tex]\mu = 3.51[/tex]
