6. (40pt) Let $x$ be a binary constant which we want to measure accurately. Suppose $x$ is measure
n times. However, there exists a noise while measuring $x$. Hence, the $i$-th measurement $Y_i$ is
given as $Y_i = x + N_i$, where $N_i$ is a random noise with zero mean and variance 10. Assume
that $Y_i$ are independent and identically distributed.
(a) (5pt) Write down the expression of sample mean of {$Y_1, Y_2, \dots, Y_n$} represented as $M_n$.
(b) (5pt) Find the mean of $M_n$.
(c) (15pt) Find the variance of $M_n$.
(d) (25pt) What is the range of $n$ such that the difference between sample mean of {$Y_1, Y_2, \dots, Y_n$}
and $x$ is less than 2 with at least probability 0.95?
