Skewness of Exponential Distribution

Theorem
Let $X$ be a continuous random variable of the exponential distribution with parameter $\beta$ for some $\beta \in \R_{> 0}$

Then the skewness $\gamma_1$ of $X$ is equal to $2$.

Proof
From Skewness in terms of Non-Central Moments, we have:


 * $\gamma_1 = \dfrac {\expect {X^3} - 3 \mu \sigma^2 - \mu^3} {\sigma^3}$

where:
 * $\mu$ is the expectation of $X$.
 * $\sigma$ is the standard deviation of $X$.

By Expectation of Exponential Distribution we have:


 * $\mu = \beta$

By Variance of Exponential Distribution we have:


 * $\sigma = \beta$

By Raw Moment of Exponential Distribution we also have:


 * $\expect {X^3} = 3! \beta^3 = 6 \beta^3$

So: