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?

Question

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?

Quizplus · Accepted Answer

The Answer of 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?

For This Question You May Assume That Linear Combinations of Normal