What formula represents the variance of a uniform distribution?

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What formula represents the variance of a uniform distribution?

The variance of a uniform distribution, specifically in the case of a continuous uniform distribution defined on the interval [a, b], is given by the formula \( \text{var}(X) = \frac{(b-a)^2}{12} \). This formula arises from the properties of the uniform distribution, where every value between a and b is equally likely, thus making the distribution symmetric. The derivation of the variance involves calculating the expected value of the squared outcomes and subtracting the square of the expected value of the outcomes. The variance measures the spread of the distribution; in the case of a uniform distribution, this spread is determined solely by the interval width, which is (b-a). Because the variance accounts for the square of this width and divides by 12, it results in a consistent measure of dispersion for all uniform distributions. Choosing the correct answer aligns with the fundamental characteristics of uniform distributions in probability theory, providing insight into how outcomes are distributed across the defined interval. The other options do not represent the variance correctly, as they either involve incorrect algebraic expressions or disregard the relationship between the width of the interval and the measure of variance.

The variance of a uniform distribution, specifically in the case of a continuous uniform distribution defined on the interval [a, b], is given by the formula ( \text{var}(X) = \frac{(b-a)^2}{12} ). This formula arises from the properties of the uniform distribution, where every value between a and b is equally likely, thus making the distribution symmetric.

The derivation of the variance involves calculating the expected value of the squared outcomes and subtracting the square of the expected value of the outcomes. The variance measures the spread of the distribution; in the case of a uniform distribution, this spread is determined solely by the interval width, which is (b-a). Because the variance accounts for the square of this width and divides by 12, it results in a consistent measure of dispersion for all uniform distributions.

Choosing the correct answer aligns with the fundamental characteristics of uniform distributions in probability theory, providing insight into how outcomes are distributed across the defined interval. The other options do not represent the variance correctly, as they either involve incorrect algebraic expressions or disregard the relationship between the width of the interval and the measure of variance.

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