$\text{Description for the following question:}$
The lifespan of a battery in a car follows Gamma distribution with probability density function
$$f(x)=\frac{\beta^\alpha }{\Gamma(\alpha) } e^{-\beta x}x^{\alpha -1}, 0<x< \infty ,$$
where $\alpha >0$ and $\beta >0$. The mean and variance of a Gamma distribution are $\mathbb E(X)=\frac{\alpha}{\beta}$ and $\mathbb V(X)=\frac{\alpha}{\beta^2 } $ respectively. From historical data the mean and variance of the lifespan of a battery are estimated as 4 years and 2 years respectively.
Which of the following statements are correct?
- $\alpha =16$ and $\beta =4$
- $\alpha =8$ and $\beta =2$
- $\mathbb E(X^2 )= \frac {\alpha}{\beta}(\frac {1+\alpha}{\beta })$
- $\mathbb E(X^2 )=18$
