Question

In a certain Poisson frequency distribution the frequency corresponding to 3 successes is
one-third the frequency corresponding to 4 successes. Find its mean and standard deviation.

Answer 1

$P(X=3)=\frac{1}{3}P(X=4)\\ e^{-\lambda}\frac{\lambda^3}{3!}=\frac{1}{3}e^{-\lambda}\frac{\lambda^4}{4!}\\ \lambda=\frac{3*4!}{3!}=12\\ Mean=Variance=\lambda=12\\ Standard\; deviation=\sqrt{12}=2\sqrt{3}$