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:

\(\ds \gamma_1\)

\(=\)

\(\ds \frac {6 \beta^3 - 3\beta^3 - \beta^3} {\beta^3}\)

\(\ds \)

\(=\)

\(\ds \frac {2 \beta^3} {\beta^3}\)

\(\ds \)

\(=\)

\(\ds 2\)

$\blacksquare$