Explanation of Standard Deviation of a Poisson Variate

Mean and Variance of a Poisson Distribution
- In a Poisson distribution, the mean and variance are equal and are denoted by λ (lambda).
- Therefore, the mean and variance of a Poisson variate X is λ.

Relationship between Variance and Standard Deviation
- The standard deviation of a random variable is the square root of its variance.
- In this case, if the standard deviation of a Poisson variate X is 2, then the variance of X is 4 (since 2^2 = 4).

Standard Deviation Formula
- The standard deviation of a Poisson distribution is the square root of its mean.
- Therefore, in this case, the standard deviation of X, which is 2, is the square root of its mean which is 4.
- Mathematically, this can be represented as σ = √λ = 2.

Interpretation
- The standard deviation of 2 indicates the spread or dispersion of the Poisson variate X around its mean.
- A larger standard deviation suggests that the values of X are more spread out from the mean, while a smaller standard deviation indicates that the values are closer to the mean.
By understanding the relationship between the mean, variance, and standard deviation of a Poisson distribution, we can interpret the significance of a standard deviation of 2 in this context.