Start at a point (0,1) in a rectangular coordinate system, choose an angle and draw the line to cross the x-axis at X. The range for X is . Find the pdf of X.
I picked the angle and now have no idea what to do.
What an odd question. Anyway, you do not seem to have a useful idea. Let's see if I can make some sense of it...
I believe the intent is to choose ANY angle, not a specific one, but ALL of them in the given range. Perhaps examples would be useful.
P(X<0) = 1/2 and this corresponds to
P(X<-1) = 1/4 and this corresponds to
P(X<1) = 3/4 and this corresponds to
Essentially, what I am trying to define is a mapping from a uniform distribution on [-pi/2,pi/2] to the entire x-axis. Sounds like it might be related to a tangent, perhaps.
It reduces to the pdf of a standard Cauchy distribution. Angle OQP is in the figure and OP=x.
The pdf of can be written as: , when .
Also, . Then the cdf of X can be given by:
, , where is the cdf of
So the pdf of X is given by:
, where is the pdf of
=
= which is a Standard Cauchy pdf.