proving pdf's
 

ahh, statistics! the joy it has brought... up until now.

i have a (uber-late) midsemester late this week, and i'm going alright, except for the proof-type questions that he said there will be one of on there, i have no idea even how to start them, i've never come across them until now.

can anyone atleast get me started?

If i have W = arctan(T/sqrt(k)) where T =d= t(k) (t-distribution, k degrees of freedom)

show that W has pdf g(w) = c(k) cos^(k-1) (w)

(where c(k) has c subscript k, and where (w) is the argument of cos^(k-1)

Well, with questions like these, which are transformations, you often do something like this:
You know the t-distribution, I assume, so you isolate W for T (rearrange to get T). Since T is t-distro, you apply the t-distro pdf to this T, noting that T is a function of W. In this case, you want to find Pr(W<w) which is equal to G(w)=Pr(function of T). Now you simply differentiate G(w) to get the pdf of W, g(w).

Another method which may work for you is to isolate W for T, then plug this new T into the formula

g(w) = f(t) * abs(dt/dw), where f(t) is the t-distro pdf for the variable T=t.

Sorry I solved it, it was a cool problem though.

The pdf of the t-distribution is

<img src="http://openset.org/images/12.jpg" style="height:97px;width:413px" />

Let's define a function,

<img src="http://openset.org/images/13.jpg" style="height:84px;width:200px" />

Now we use my second method above to get

<img src="http://openset.org/images/14.jpg" style="height:74px;width:541px" />

Plugging in we see that

<img src="http://openset.org/images/18.jpg" style="height:175px;width:425px" />

Using a trig identity we have

<img src="http://openset.org/images/16.jpg" style="height:31px;width:265px" />

Now just simplify.

<img src="http://openset.org/images/17.jpg" style="height:67px;width:641px" />

ahhhh... its all coming back to me now! this is probability stuff from 1st semester, method of transformations!

the c(k) is what got me - i got it confused with order statistics, and couldnt see how in hell i could work it in there.

you, my good sir, are a champion!

back to revision!