Final answer:
The possible values for the variance when it is equal to the standard deviation are either 0 or 1. This is based on the definition and properties of standard deviation and variance in a probability distribution.
Step-by-step explanation:
When considering a distribution where the standard deviation is equal to the variance, we must recognize that the only possible values for the variance are 0 or 1. This is because the standard deviation is a measure of dispersion and is always positive or zero, and is defined as the square root of the variance. Therefore, the only non-negative variance that is equal to its own square root (standard deviation) is 0 or 1.
Let's consider the formula for variance, σ² = Σ (x − μ) ² P(x), where σ² symbolizes the variance and x represents values of the random variable with the mean μ, and P(x) represents the corresponding probability. Following the property that variance must be the square of standard deviation, if σ is equal to σ², only the values 0 or 1 satisfy this condition.