1. ## expectaion^2 and variance^2

I need help solving these:

$E\left[\left(\frac{\overline{x}-\mu}{\sigma/6}\right)^2 \right]$

$Var\left[\left(\frac{\overline{x}-\mu}{\sigma/6}\right)^2\right]$

(btw, what the code to make both delimiters tall.)

:-)

Thanks.

2. Originally Posted by yvonnehr
I need help solving these:

$E\left[\left(\frac{\overline{x}-\mu}{\sigma/6}\right)^2 \right]$

$Var\left[\left(\frac{\overline{x}-\mu}{\sigma/6}\right)^2\right]$

(btw, what the code to make both delimiters tall.)

:-)

Thanks.
Expand the expression inside the expectation and then use known properties of expectation to simplify (mainly linearity)

For the variance re write as an expectation and repeat the above process.

\left( \right)

CB

3. BY any chance, are you sampling from a normal population with n=36?
If so, the rv inside the square root is a st normal, which makes your life easier.
Even if n isn't 36, normality makes this a lot easier.

4. ## n=36, Thanks!

Yes! n=36.

Thanks guys! I'll work on this and see how it goes.

:-)

5. If the 36 observations are iid Normals, then

$\frac{\overline{X}-\mu}{\sigma/6}\sim N(0,1)$

and you just need the distribution of a st normal squared, which is a Chi-Square with 1 df.
And the mean and variance of a chi-square is trivial.

6. ## Z^2 ~ X^2 df 1 - Why?

Originally Posted by matheagle
If the 36 observations are iid Normals, then

$\frac{\overline{X}-\mu}{\sigma/6}\sim N(0,1)$

and you just need the distribution of a st normal squared, which is a Chi-Square with 1 df.
And the mean and variance of a chi-square is trivial.
Matheagle,
So I finally found a reference that tells me that the distribution of Z^2 is Chi-squared with df 1. But I didn't find anything that shows me how or why that works.

Will you tell me or point me to somewhere I can see a proof or some explanation? (You must pardon me but I am a math major and like to see how things work.)

Thanks a bunch for your help!

7. All that work is a waste of time if $X_1,...,X_{36}$ are NOT from a normal population.
I asked if they were and I don't see a response.

8. ## Response to ? about normal pop.

Originally Posted by matheagle
All that work is a waste of time if $X_1,...,X_{36}$ are NOT from a normal population.
I asked if they were and I don't see a response.
Yes, it is from a normal population.

9. all this is in many books, including walpole.
Use MGFs to prove that the linear combination of normals is normal
and then use that 2-1 transformation to prove that the square of a st normal is a chi-square.
I bet I can find it online.
I can prove it without looking it up, but I want to make dinner and grade exams tonight.

10. ## Thanks for the proof outline.

Originally Posted by matheagle
all this is in many books, including walpole.
Use MGFs to prove that the linear combination of normals is normal
and then use that 2-1 transformation to prove that the square of a st normal is a chi-square.
I bet I can find it online.
I can prove it without looking it up, but I want to make dinner and grade exams tonight.
Okay, great. I can try it/find it with the outline you have given. Thanks!

Have a good dinner.

11. I couldn't find it, but I'll start it for you.

Let z be a st normal and $X=Z^2$

$F_X(x)=P(X\le x)=P(Z^2\le x)=P(-\sqrt{x}\le Z\le \sqrt{x} )=F_Z(\sqrt{x})-F_Z(-\sqrt{x})$

NOW differentiate wrt x and obtain the realtionship between the two densities and then plug in.

${d\over dx}\Biggl(F_X(x)=F_Z(\sqrt{x})-F_Z(-\sqrt{x})\Biggr)$

$f_X(x)=f_Z(\sqrt{x}) {1\over 2\sqrt{x}} -f_Z(-\sqrt{x}){-1\over 2\sqrt{x}}$

$=\biggl(f_Z(\sqrt{x})+f_Z(-\sqrt{x})\biggr){1\over 2\sqrt{x}}$

NOW plug in.

12. ## Ha ha!

Originally Posted by matheagle
I couldn't find it, but I'll start it for you.
Let z be a st normal and $X=Z^2$
Oh, you are real funny! That's a good one!!! The wit of a teacher is unmistakable.

13. Originally Posted by yvonnehr
Matheagle,
So I finally found a reference that tells me that the distribution of Z^2 is Chi-squared with df 1. But I didn't find anything that shows me how or why that works.

Will you tell me or point me to somewhere I can see a proof or some explanation? (You must pardon me but I am a math major and like to see how things work.)

Thanks a bunch for your help!
See post #7 in this thread: http://www.mathhelpforum.com/math-he...tml#post119542