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The Poisson distribution describes the probability of a given number of events occurring within a fixed interval of time or space. It is characterized by the parameter \( \mu \), which represents the average number of events. The average value of a random variable in a Poisson distribution is also called the mean. To find this average value, denote the random variable by \( X \) and use its probability mass function (pmf): \( P(X = k) = \frac{\mu^k e^{-\mu}}{k!} \).
The expected value, or mean, of the Poisson distribution can be found by: \( E(X) = \sum_{n=0}^{\infty} n \frac{\mu^n e^{-\mu}}{n!} \).
Through careful computation and recognizing how the sums interact with the exponential function, one finds that \(E(X) = \mu\).
This result makes intuitive sense because in a Poisson process with average rate \(\mu\), \(\mu\) is the expected number of occurrences.

The standard deviation is a measure of the amount of variation or dispersion in a set of values. For the Poisson distribution, the variance and the mean are equal. This can be derived from properties of the Poisson distribution:
The variance of a Poisson-distributed random variable X is given by: \( \sigma^2 = Var(X) = E(X^2) - (E(X))^2 = \mu \).
Therefore, the standard deviation \( \sigma \) is the square root of the variance: \( \sigma = \sqrt{\mu} \).
This indicates that as \(\mu\) increases, both the mean and the variation around that mean also increase.

The probability mass function (pmf) of a discrete random variable gives the probability that the variable is exactly equal to some value. For the Poisson distribution, the pmf is given by:
\( P(X = k) = \frac{\mu^k e^{-\mu}}{k!} \), where \( k \) is a non-negative integer.
This formula shows how the probability of observing exactly \( k \) events is determined by the mean number of events \( \mu \). The exponential function \( e^{-\mu} \) decays, ensuring that the probabilities sum to 1 over all possible values of \( k \). The factorial term \( k! \) adjusts for the number of ways \( k \) events can occur.

The exponential function \( e^x \) can be expressed as an infinite series, a key concept in finding moments of a Poisson distribution. The series for \( e^x \) is:
\( e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \).
This representation is powerful because it allows us to manipulate and differentiate the function to obtain necessary identities. For example, differentiating \( e^x \) gives it back, as \( \frac{d}{dx}(e^x) = e^x \). Multiplying both sides by \( x \) lets us connect the series to \( \mu \) in the Poisson distribution.

In statistics, the mean is the average value of a random variable, while the variance measures the spread of its values. For the Poisson distribution, calculating the mean and variance involves using its pmf and properties of the exponential function series. The mean \( \mu \) is given by the expected value: \( E(X) = \sum_{n=0}^{\infty} n \frac{\mu^n e^{-\mu}}{n!} = \mu \).
The variance is: \( Var(X) = \sigma^2 = E(X^2) - (E(X))^2 = \mu \)
Consequently, these results confirm that both the mean and the variance for the Poisson distribution are equal to \( \mu \).