# density function question..

• Feb 2nd 2010, 02:57 AM
transgalactic
density function question..
$Y\sim U(-2\pi,2\pi)$
find the density function of z=tan(Y)
?

X~U(0,1)
find the density function of W=a+bx
the solution is
W~U(a,a+b)

how to solve the first question ??
• Feb 2nd 2010, 05:32 PM
matheagle
READ below, I think the orginal density is WRONG.

USE $f_Z(z)=f_Y(y)\left| {dy\over dz}\right|$ which is easy to derive.

It's the derivatives of the CDFs, it's just calc 1.

So, $f_Y(y)={1\over 4\pi}$ on $(-2\pi, 2\pi)$

and $y=\arctan z$ giving you ${dy\over dz}={1\over 1+z^2}$

Hence $f_Z(z)={1\over 4\pi(1+z^2)}$

I THINK there is a problem here, we should be getting a Cauchy.

I think the original density should be a uniform on $(-\pi/2, \pi/2)$ because tangent isn't defined at $\pi/2$
• Feb 2nd 2010, 05:42 PM
matheagle
The second one is obvious.

$f_X(x)=1$ on 0<x<1.

If you let W=a+bX then W exists on (a,a+b), by letting X=0 and X=1.
Next, I will show that W is uniform on that interval.

USE $f_W(w)=f_X(x)\left| {dx\over dw}\right|$

With w=a+bx we have ${dw\over dx}=b$

So, USE $f_W(w)=(1)\left| {1\over b}\right|={1\over b}$ IF b>0.

which is a uniform density.