If λ is the parameter of a Poisson distribution, what does the variance of this distribution equal?

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If λ is the parameter of a Poisson distribution, what does the variance of this distribution equal?

In a Poisson distribution, the parameter λ represents both the mean and the variance of the distribution. Therefore, if we denote the variance of a Poisson random variable as Var(X), it is equal to λ. This is a fundamental property of the Poisson distribution, where the occurrences of events in a fixed interval of time or space occur with a known constant mean rate and independently of the time since the last event. In essence, for any Poisson-distributed random variable, as the average rate of occurrences (λ) increases, so does the variability (or variance) around that mean. Consequently, the variance being equal to λ illustrates that the greater the average number of events, the larger the dispersion around that average. The other choices do not align with the properties of a Poisson distribution. For example, the value of 1 is not universally applicable to all Poisson distributions, and the choice of 2λ or 0 does not hold true since they do not correspond to the established relationship between the mean and the variance in this specific statistical distribution. Thus, the variance being equal to λ is the correct and definitive property of the Poisson distribution.

In a Poisson distribution, the parameter λ represents both the mean and the variance of the distribution. Therefore, if we denote the variance of a Poisson random variable as Var(X), it is equal to λ. This is a fundamental property of the Poisson distribution, where the occurrences of events in a fixed interval of time or space occur with a known constant mean rate and independently of the time since the last event.

In essence, for any Poisson-distributed random variable, as the average rate of occurrences (λ) increases, so does the variability (or variance) around that mean. Consequently, the variance being equal to λ illustrates that the greater the average number of events, the larger the dispersion around that average.

The other choices do not align with the properties of a Poisson distribution. For example, the value of 1 is not universally applicable to all Poisson distributions, and the choice of 2λ or 0 does not hold true since they do not correspond to the established relationship between the mean and the variance in this specific statistical distribution. Thus, the variance being equal to λ is the correct and definitive property of the Poisson distribution.

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