Suppose that $X_1, X_2, ..., X_n$ is a random sample representing the number of defective elec-\tronic components, out of $k$ components, that may be manufactured by a certain company on\any of $n$ randomly chosen days. Furthermore, suppose that the distribution of the number of\defective components ($X$), out of $k$ components, that may be manufactured by the company\on any day is binomial($k$, $\theta$) with probability density function:\
$\begin{cases}
\binom{k}{x} \theta^x (1-\theta)^{k-x} & \text{if } x = 0, 1, ..., k,\\
0 & \text{elsewhere.}\
\end{cases}$\
(a) Write down $E(X)$ and $V(X)$.\\(2)\
(b) Assume that $\theta = \frac{1}{2}$. Derive the method of moments estimator of $k$.\\(3)\
(c) Assume that $k = 10$. Derive an equation whose solution is the method of moments\estimator of $\theta$. Hint: $V(X) = E(X^2) - [E(X)]^2$.\\(5)
