Hi all,
I was reading through my textbook and I could not solve any of these questions.
2.24 Suppose Y_{i} 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 \(\displaystyle t_{n1}\)
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.
I was reading through my textbook and I could not solve any of these questions.
2.24 Suppose Y_{i} 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 \(\displaystyle t_{n1}\)
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.
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