finding a constant for t distribution

**Fred1956** · November 26th 2009, 06:57 PM

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

**matheagle** · November 26th 2009, 09:39 PM

Let $A=X_1 + X_2$ and $B=X_3^2+X_4^2+X_5^2$

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

and $B\sim \chi^2_3$

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

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

It looks like $c=\sqrt{3/2}$

**Fred1956** · November 28th 2009, 04:10 PM

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

**matheagle** · November 28th 2009, 05:17 PM

$V(aX+b)=a^2V(X)$ but that doesn't give you normality

You should know that, if

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

which is what I used. I'm sure you covered that if you're examining t distributions.

**Fred1956** · November 28th 2009, 07:18 PM

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

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