Variance of Exponential Distribution

Theorem
Let $X$ be a discrete random variable with the exponential distribution with parameter $\beta$.

Then the variance of $X$ is:
 * $\operatorname{var} \left({X}\right) = \beta^2$

Proof
From the definition of Variance as Expectation of Square minus Square of Expectation:
 * $\operatorname{var} \left({X}\right) = E \left({X^2}\right) - \left({E \left({X}\right)}\right)^2$

From Expectation of Exponential Distribution:
 * $E \left({X}\right) = \beta$

The expectation of $X^2$ is:

Thus the variance of $X$ is: