Final answer:
The Poisson distribution is a probability distribution that gives the probability of a number of events occurring in a fixed interval of time or space. To show that it is properly normalized, we need to prove that the sum of all possible probabilities in the distribution equals 1. This can be done by applying Taylor series expansion and then summing the resulting series.
Step-by-step explanation:
The Poisson distribution is a probability distribution that gives the probability of a number of events occurring in a fixed interval of time or space, if these events happen with a known average rate and independently of the time since the last event. To show that it is properly normalized, we need to prove that the sum of all possible probabilities in the distribution equals 1. In other words, we need to show that the sum of P(X=x) for all values of x from 0 to infinity is equal to 1.
Since the Poisson distribution is defined for positive integer values of x, we can extend the sum to infinity because the terms become very small when n ≥ N. Therefore, to prove that the Poisson distribution is properly normalized, we need to show that the sum of xe^(-λ) * λ^x / x! from 0 to infinity is equal to 1.
This can be done by applying Taylor series expansion to the term e^(-λ) * λ^x / x! and then summing the resulting series. The result will be 1, which confirms that the Poisson distribution is properly normalized.