probabity of 3 chances of a event is 2/3 times the probabity of 4 chances. Find mean & SD

Question

i think it is a poisson distribution and have to use the pmf formula and this is the question

 

In a Poisson distribution, the probability of 3 occurrences of an event is 2/3 times the probability of 4 occurrences. Find the mean and standard deviation of the distribution. 

Answer 1

Po(m;x)=e^-m * m^x/x!  symbolises the Poisson probability of x events for a mean m events.

When x=3 Po(m;3)=e^-m * m^3/6 and when x=4 Po(m;4)=e^m * m^4/24.

Po(m;3)=(2/3)Po(m;4), so m^3/6=(2/3)m^4/24, 1/6=m/36 and m, the mean=36/6=6.

In a Poisson distribution the mean and variance are the same, and SD=√variance, so SD=√6=2.4495 approx.