(Solved): Suppose that the random variable X has the Gamma distribution where =2 (scale parameter) and r=4 ...

Suppose that the random variable $X$ has the Gamma distribution where $?=2$ (scale parameter) and $r=4$ (shape parameter). Using the scale and shape parameter, the expected value is calculated as $E[X]=\frac{r}{?}$, and the variance is calculated as $V[X]=\frac{r}{{?}^2}$. More information on this probability distribution can be found in Section 4.8 of the textbook. For your convenience, the PDF is provided below: $f(x)=\{\begin{array}{cc}\frac{{?}^r{x}^{r?1}{e}^{??x}}{?(r)},& x>0.\\ 0,& \text{ otherwise. }\end{array}$ a) [6 points] A random sample of $n=50$ is selected from this distribution. Calculate the expected value and variance of the sample mean. b) [ 6 points] Using the same random sample of $n=50$, calculate the probability that the sample mean is greater than, i.e., $P(\stackrel{?}{X}>2.2)$. c) [6 points] If the random variable $X$ was distributed as a continuous uniform distribution with parameters $a=1$ and $b=3$ calculate the expected value and variance of the sample mean (use the same random sample of size $n=50$ ). d) [2 points] From part (c) above, would the $P(\stackrel{?}{X}>2.2)$ be more or less than the value calculated for part (b)? Explain how you could use the expected value and the variance calculations from part (a) and (c) to answer this question.