Suppose that a random variable $X$ is distributed according to a gamma distribution with parameters $\alpha = 6$ and $\beta = 2$, i.e., $X \sim \text{Gamma}(6, 2)$.
A.) Computer the mean and variance of $X$.
$E(X) = \alpha \beta = 6\times2 = 12$
$\text{Var}(X) = \alpha \beta^{2} = 6\times2^2 = 24$
B.) Find $E(X^4)$
I believe I found the correct values for the mean and variance but I am having trouble calculating $E(X^4)$. I know $E(X^2) = \text{Var}(X) + E^2(X)$ but I have no idea what to do for $E(X^4)$. Any help would be much appreciated.
Best Answer
From wikipedia, the moments are given by the formula
$$E[X^n] = \theta^n \cdot \frac{\Gamma(n+k)}{\Gamma(k)}$$
where $\theta$ is the scale parameter and $k$ is the shape parameter.
Since you stated that you found the correct answer in the first part, you are not using the standard notation. Your $\alpha$ is the shape parameter and $\beta$ is the scale paarameter.