Proving Distribution: Suppose Y_i is distributed i.i.d. N(0, sigma^2) for i=1,2,3,...

xwy

New member
Joined
Aug 5, 2016
Messages
2
Hi all,

I was reading through my textbook and I could not solve any of these questions.



2.24 Suppose Yi is distributed i.i.d. \(\displaystyle N(0,\, \sigma^2)\) for i = 1, 2, 3,..., n.

. . . . . a. Show that \(\displaystyle E(Y_i^2\, /\, \sigma^2)\, =\, 1\)

. . . . . b. Show that \(\displaystyle W\, =\, (1/\sigma^2)\, \sum_{i=1}^n\, Y_i^2\) is distributed \(\displaystyle \chi_n^2\)

. . . . . c. Show that \(\displaystyle E(W)\, =\, n.\) [Hint: Use your answer to (a).]

. . . . . d . Show that \(\displaystyle V\, =\, Y_i\, /\, \sqrt{\dfrac{\sum_{i=1}^n\, Y_i^2}{n\, -\, 1}\,}\) is distributed \(\displaystyle t_{n-1}\)




The painful part is, I don't even know how and where to start. It'll be great if anyone of you could help me out. I've spend like hours looking through the textbook and internet for solution but to no avail.


Thank you in advance!

P.S: I've just started my term and my professor hasn't been teaching much.
 

Attachments

Last edited by a moderator:

tkhunny

Moderator
Staff member
Joined
Apr 12, 2005
Messages
9,787
I was reading through my textbook and I could not solve any of these questions.



2.24 Suppose Yi is distributed i.i.d. \(\displaystyle N(0,\, \sigma^2)\) for i = 1, 2, 3,..., n.

. . . . . a. Show that \(\displaystyle E(Y_i^2\, /\, \sigma^2)\, =\, 1\)

. . . . . b. Show that \(\displaystyle W\, =\, (1/\sigma^2)\, \sum_{i=1}^n\, Y_i^2\) is distributed \(\displaystyle \chi_n^2\)

. . . . . c. Show that \(\displaystyle E(W)\, =\, n.\) [Hint: Use your answer to (a).]

. . . . . d . Show that \(\displaystyle V\, =\, Y_i\, /\, \sqrt{\dfrac{\sum_{i=1}^n\, Y_i^2}{n\, -\, 1}\,}\) is distributed \(\displaystyle t_{n-1}\)




The painful part is, I don't even know how and where to start. It'll be great if anyone of you could help me out. I've spend like hours looking through the textbook and internet for solution but to no avail.
You should select a methods of comparison. Find the fundamental equation of the starting distribution and the terminal distribution. Do they have the same moments? Can you transform one integral into the other? Give it a go!
 
Last edited by a moderator:
Top