What is the variance of a Poisson distribution represented by?

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What is the variance of a Poisson distribution represented by?

In a Poisson distribution, the variance is a key characteristic that reflects the spread of the distribution around its mean. For a Poisson random variable \( X \) with parameter \( \lambda \), both the mean and the variance are equal to \( \lambda \). This means that if the average number of events (mean) occurring in a fixed interval is \( \lambda \), the variability of the number of events around this average (variance) is also \( \lambda \). The reason this property of equality between the mean and the variance stands out is that the Poisson distribution is often used to model the number of occurrences of an event in a fixed interval, where these occurrences happen independently of one another. This unique relationship is part of the defining features of the Poisson distribution, making \( \lambda \) representative of both the central tendency and the dispersion. This characteristic helps in practical applications where understanding both the expected outcome and the variability of that outcome is crucial, such as in fields like insurance or queueing theory. The fact that the variance equals \( \lambda \) is what makes option B the correct choice.

In a Poisson distribution, the variance is a key characteristic that reflects the spread of the distribution around its mean. For a Poisson random variable ( X ) with parameter ( \lambda ), both the mean and the variance are equal to ( \lambda ). This means that if the average number of events (mean) occurring in a fixed interval is ( \lambda ), the variability of the number of events around this average (variance) is also ( \lambda ).

The reason this property of equality between the mean and the variance stands out is that the Poisson distribution is often used to model the number of occurrences of an event in a fixed interval, where these occurrences happen independently of one another. This unique relationship is part of the defining features of the Poisson distribution, making ( \lambda ) representative of both the central tendency and the dispersion.

This characteristic helps in practical applications where understanding both the expected outcome and the variability of that outcome is crucial, such as in fields like insurance or queueing theory. The fact that the variance equals ( \lambda ) is what makes option B the correct choice.

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