Question

For a Poisson distributed random variable X, P(X = 4) = P(X = 5).Find the mean
and variance of the distribution.

Answer 1

$P(X=4)=P(X=5) -> \lambda^{4}e^{-\lambda}/4!=\lambda^{5}e^{-\lambda}/5! -> \lambda=5.\\$

So, $mean=variance=\lambda=5.$