A discrete random variable X is called a truncated Poisson...

A discrete random variable $X$ is called a truncated Poisson random variable if its mass points are $0,1,2, \ldots, N$ and its probability mass function $p(\cdot)$ is given by $p(k)=C e^{-\alpha} \alpha^k / k !, k=0,1,2, \ldots, N$. What is the value of the constant $C$ ?

Key Concepts

Discrete Random Variable

A discrete random variable is one that takes on a countable number of distinct values. In probability theory, these variables are defined by their probability mass functions (PMFs), which assign probabilities to each potential outcome in a way that the sum of all probabilities equals one.

Probability Mass Function (PMF)

A probability mass function is a function that gives the probability that a discrete random variable is exactly equal to a specific value. It is a fundamental tool in statistics for describing the distribution of discrete outcomes, ensuring that probabilities across all outcomes sum up to one.

Poisson Distribution

The Poisson distribution is a probability distribution that models the number of times an event occurs in a fixed interval of time or space, given that these events occur with a known constant mean rate and independently of the time since the last event. It is typically defined over the non-negative integers with a PMF proportional to e^(-alpha) * (alpha^k) / k!.

Truncated Distribution

A truncated distribution is derived from an existing probability distribution by restricting its domain to a subset of the original range. In the context of the truncated Poisson distribution, the variable is limited to a finite set of values, namely from 0 to N, and the original probabilities are renormalized to sum up to one over this restricted range.

Normalization Constant

The normalization constant in a probability distribution is a scaling factor that ensures the sum (or integral) of probabilities across all outcomes equals one. In the case of a truncated distribution, this constant is determined by summing the unnormalized probabilities over the allowed range and then taking the reciprocal of that sum, effectively reweighting the probabilities.

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