For This Question You May Assume That Linear Combinations of Normal

question 5

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?

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