Uniform Distribution 3: Variance

What is the variance of a uniform random variable with pdf $$ f(x) = 1 $$ for $0 \leq x \leq 1$ and zero otherwise?

Answer

For a uniform distribution on $$[0, 1]$$ , the variance $\text{Var}(X)$ is calculated as:

\text{Var}(X) = \int_{0}^{1} x^2 \, dx - \left( \int_{0}^{1} x \, dx \right)^2 = \frac{1}{3} - \left( \frac{1}{2} \right)^2 = \frac{1}{12}

Therefore, the variance of the uniform random variable is $\frac{1}{12}$ .

Notes and comments

Comment: For a general uniform distribution on $$[a, b]$$ , the variance is:

\text{Var}(X) = \frac{(b - a)^2}{12}