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Let X be a Poisson random variable with parameter λ. Use...

Let $X$ be a Poisson random variable with parameter $\lambda .$ Use Exercise 23 in Section 11.5 to show that the probability that $X$ is an even integer (including 0 ) is $e^{-\lambda}$ cosh $\lambda$.

Key Concepts

Poisson Distribution

The Poisson distribution characterizes the probability of a given number of events occurring in a fixed interval of time or space when these events happen with a known constant mean rate and independently of the time since the last event. Its probability mass function is defined as P(X = k) = e^(-?)?^k/k!, where ? is the average rate of occurrence. This distribution is fundamental in probability theory and statistics when modeling count data events.

Probability Generating Function (PGF)

The probability generating function is a powerful tool that encodes the probability mass function of a discrete random variable. For the Poisson distribution, the PGF is given by G(t) = exp(?(t - 1)), which facilitates the computation of probabilities for events defined by specific conditions, such as the probability of the variable being even or odd. The PGF serves as a handy method to derive moments and manipulate series representations in probability theory.

Parity and Series Decomposition

When determining probabilities over a subset of outcomes, such as even or odd values, series decomposition techniques come into play. By summing the terms of the probability mass function over even indices, one can separate the series into two parts based on parity. This approach simplifies the calculation of the probability of the event by matching series representations with known functions, demonstrating the connection between combinatorial series and probability.

Hyperbolic Cosine Function

The hyperbolic cosine function, cosh(x), is defined as (e^x + e^(-x))/2 and has a series expansion given by cosh(x) = ? (x^(2n))/(2n)! over n from 0 to infinity. In the context of probability, this series naturally appears when summing even-powered terms, as seen in the computation of the probability that a Poisson random variable takes on even values. Recognizing this connection between the series expansion and the hyperbolic cosine function allows for a concise expression of the probability result.

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