For This Question You May Assume That Linear Combinations of Normal

Essay

For this question you may assume that linear combinations of normal variates are themselves normally distributed. Let a, b, and c be non-zero constants.
(a)X and Y are independently distributed as N(a, σ²). What is the distribution of (bX+cY)?
(b)If X₁,..., X_n are distributed i.i.d. as N(a, $\sigma _ { X } ^ { 2 }$ ), what is the distribution of $\frac { 1 } { n }$ $\sum _ { i = 1 } ^ { n } X _ { i }$ ?
(c)Draw this distribution for different values of n. What is the asymptotic distribution of this statistic?
(d)Comment on the relationship between your diagram and the concept of consistency.
(e)Let $\bar { X }$ = $\frac { 1 } { n }$ $\sum _ { i = 1 } ^ { n } X _ { i }$ What is the distribution of $\sqrt { n }$ ( $\bar { X }$ - a)? Does your answer depend on n?