Limit of rescaled Binomials
Let Xn be a binomial random variable with parameters n and p = lambda/n, where lambda is a fixed positive number.
Let k be a natural number. As n tends to infinity, the probability mass function P(Xn = k) converges to a number that only depends on lambda and k. What is P(Xn = k) in the limit as n tends to infinity?
To find the limit of P(Xn = k) as n tends to infinity, we can use the concept of the Poisson approximation to the binomial distribution.
In the limit as n tends to infinity, the binomial distribution approaches a Poisson distribution with parameter lambda. The probability mass function of a Poisson distribution is given by P(X = k) = (e^(-lambda) * lambda^k) / k!, where e is the base of the natural logarithm.
Therefore, in the limit as n tends to infinity, P(Xn = k) converges to P(X = k) = (e^(-lambda) * lambda^k) / k!. The probability only depends on lambda and k.