Poisson Distribution Standard Deviation Explained

The question asks to determine the standard deviation for a Poisson distribution, given that its mean is denoted by the symbol $\mu$. Understanding the fundamental properties of a Poisson distribution is crucial to answer this question.

Understanding the Poisson Distribution

A Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. While the question mentions "tossing of a biased coin," this is typically modeled by a binomial distribution; however, for the purpose of this question, we are specifically asked about the properties of a Poisson distribution itself.

Key Parameters of a Poisson Distribution

For any Poisson distribution, there is a very important relationship between its mean and its variance. Let's look at these parameters:

• Mean ($\mu$): The average number of events in the given interval. In a Poisson distribution, the mean is typically denoted by $\lambda$ or, as given in this question, $\mu$.
• Variance ($\sigma^2$): A measure of how spread out the numbers are from the mean. For a Poisson distribution, a unique property is that its variance is exactly equal to its mean.
• Standard Deviation ($\sigma$): The square root of the variance. It indicates the typical distance of data points from the mean.

Calculating Standard Deviation for Poisson Distribution

Given the properties mentioned above, we can derive the standard deviation.

• Let $X$ be a random variable following a Poisson distribution.
• The mean of a Poisson distribution is given as $\mu$. So, $E(X) = \mu$.
• A key property of the Poisson distribution is that its variance is equal to its mean. Therefore, $Var(X) = \mu$.
• The standard deviation ($\sigma$) is defined as the square root of the variance.
• Thus, $\sigma = \sqrt{Var(X)}$.
• Substituting the variance, we get $\sigma = \sqrt{\mu}$.

Therefore, if the mean of a Poisson distribution is $\mu$, its standard deviation will be $\sqrt{\mu}$.

Summary of Poisson Distribution Properties

Based on this analysis, the standard deviation for a Poisson distribution with mean $\mu$ is $\sqrt{\mu}$.