Solve for q Value in F Distribution with Degrees of Freedom 1 & 2

[rhyno89]

Homework Statement

If X and Y are independent chi square random variables with degrees of freedom of 1 and 2 respectively, then W = 3X/Y with P(W >= q) .25 holds for what q value

Homework Equations

The Attempt at a Solution

So I know that the result of two chi square distributions with one being divided into another results in an F distribution. Where I am getting really confused is with the factor of 3 before the X random variable. My guess was to set it up as P(3F1,2 >= q) = .25 For the time being I just figured it out as 1-P(F1,2 <= q) since the values on the table are cummulative to find the q value. At this point I jus multiplied it by 3 to account for the multiplicative factor but that seems too easy and does not make much since.

On a side note how do I know whether or not the F statistic is 3X/Y or Y/3X

Thanks a lot fellas

 

[lanedance]

why not start by considering the RV, Z = 3X? Then once you get your head around Z, you can consider W = Z/Y

 

[rhyno89]

since there is an additive property with independent chi square distributions...could i treat the 3X as simply X+X+X and get a chi square random variable with 3 degrees of freedom?

 

[earlh]

since there is an additive property with independent chi square distributions...could i treat the 3X as simply X+X+X and get a chi square random variable with 3 degrees of freedom?

Do you actually have 3 degrees of freedom?

 

[rhyno89]

Ok i understand that part...i can't use the additive property because I am not adding three different independent random variables, I am merely multiplying the one that I do have by 3.

Since I can't use 3 df i have to still use the 1 and 2 respectively. It seems that a logical attempt would be to actually write out the pdf of a chi square df 1 and multiply it by 3 and divide it by the pdf of the chi square of df 2 would be one way to get the F distribution and solve it. I did not want to go this route because it seemed that there was a theorem or definition that would deal with a chi square RV transformation

 

[lanedance]

that sounds like a good idea, so how about this... consider the variable Z = (X/1)/(Y/2) = X/(2Y), then Z has an F(1,2) distribution, see following
http://en.wikipedia.org/wiki/F-distribution

then W = 6 Z = 3X/Y

so P(W>=q) = 0.25 is equivalent to P(Z>=q/6)

 

[rhyno89]

ok thanks that makes a lot of sense... i think that is similar to what i initially tried to do and just got confused along the way...

my only question about that is what property of the F distribution results in being able to rewrite X with df 1 / Y with df 2 as X/2Y and since u can do that is that saying that similarly if Y had 4 df u could rewrite it as X/4Y?

 

[lanedance]

have a look at the wiki page, if you have two chi square variables with degrees of freedom

X - dfX
Y - dfY

The following RV will have a an F(dfX, dfY) distribution
Z = (X/dfX)/(Y/dfY)