finding a constant for t distribution

• Nov 26th 2009, 06:57 PM
Fred1956
finding a constant for t distribution
I have a homework problem asking me to find the constant c that will make the statistic have a t-distribution. The rv X1...X5 are iid with a standard normal distribution.

The statistic is:

T = c(X1 + X2)/sqrt (X3^2+X4^2+X5^2)

I see that the denominator looks like a Chi-Square with 3df and the numerator could be the summation of X. I know I need X-N(0,1) in numerator and U-Chi(rdf) in denominator for a t-distribution t(rdf). I want to square both sides to start but that looks more like a F-distribution. Any hints would be appreciated.

Thanks

Fred1956
• Nov 26th 2009, 09:39 PM
matheagle
Let $\displaystyle A=X_1 + X_2$ and $\displaystyle B=X_3^2+X_4^2+X_5^2$

$\displaystyle A\sim N(0,2)$ hence $\displaystyle {A\over \sqrt{2}}\sim N(0,1)$

and $\displaystyle B\sim \chi^2_3$

THUS $\displaystyle {{A\over\sqrt{2}}\over\sqrt{{B\over3}}}\sim t_3$

You could have squared everything and obtain an $\displaystyle F_{1,3}$

It looks like $\displaystyle c=\sqrt{3/2}$
• Nov 28th 2009, 04:10 PM
Fred1956
finding a constant for t distribution
I see you get the A~N(0,2) by adding summing the variances for X1 and X2. But how to you get the A/sqrt2. I would assume from another rule for variance but am not sure what.

Thanks

Fred1956
• Nov 28th 2009, 05:17 PM
matheagle
$\displaystyle V(aX+b)=a^2V(X)$ but that doesn't give you normality

You should know that, if

$\displaystyle X\sim N(\mu,\sigma^2)$ then $\displaystyle {X-\mu\over\sigma}\sim N(0,1)$

which is what I used. I'm sure you covered that if you're examining t distributions.
• Nov 28th 2009, 07:18 PM
Fred1956
finding a constant for t distribution
Thanks

I am trying to understand the math operations that apply. Do you treat the tilde (~) like an equal sign and then you can perform math operations across it. It looks like you subtrated mu from both sides and then divided by sigma squared but you ended up with sigma instead of sigma squared under the X-mu. Can you clearify the math operations for me.

Fred1956