Answer:
[tex]P(X=x)=\frac{\frac{1}{3}^{x}*e^{-\frac{1}{3}} }{x!}[/tex]
Step-by-step explanation:
According to the probabilistic relationship between the exponential distribution and the Poisson distribution, which expresses that if the time between events is exponential with mean m (rate L=1/m) then the number of events of a t time is Poisson with a L*t parameter. Therefore, the probability mass function is given by,
[tex]P(X=x)=\frac{L^{x}*e^{-L} }{x!}[/tex]
Where,
[tex]L=\frac{1}{m}[/tex]
m: mean
In this case as we have that the mean is 3 (that is m=3), then the probability mass function of X is:
[tex]P(X=x)=\frac{\frac{1}{3}^{x}*e^{-\frac{1}{3}} }{x!}[/tex]