A Poisson random variable is a discrete random variable that represents the number of occurrences of a certain event in a fixed interval of time or space. It is often used to model rare events and is characterized by a single parameter, λ, which represents the average rate of occurrence.
Two random variables are considered independent if the occurrence of one does not affect the probability of the other occurring. In other words, the outcome of one variable has no influence on the outcome of the other.
If X and Y are two Poisson random variables with parameters λ1 and λ2, respectively, they are considered independent if the probability of X=k and Y=m occurring simultaneously is equal to the product of their individual probabilities, P(X=k) * P(Y=m).
The probability mass function of a Poisson random variable is given by P(X=k) = (e^-λ * λ^k) / k!, where λ is the parameter and k is the number of occurrences.
The mean and variance of a Poisson random variable are both equal to the parameter λ. This can be calculated using the formula E(X) = Var(X) = λ. This means that the mean and variance are equal and can be used interchangeably.