 Register Blogs Wiki FAQ Calendar Search Today's Posts Mark Forums Read
 FlashChat Actuarial Discussion Preliminary Exams CAS/SOA Exams Cyberchat Around the World Suggestions

 DW Simpson Global Actuarial & Analytics Recruitment Download our Actuarial Salary Survey now with state-by-state salary information!

 Probability Old Exam P Forum

#1
 jchaney Member CAS Join Date: Oct 2014 College: Univ. of Washington Posts: 96 Finan 38.7

$f(x) = \frac{1}{2\pi}$
$y = cos x$

find pdf of y.

1. arccos y = x
2. d/dx arcos y = - 1/((1-x^2)^.5)
3. (1/2pi)*| - 1/((1-x^2)^.5)| = 1/(2pi*(1-x^2)^.5)
But, solution is 1/(pi*(1-x^2)^.5)
Where did I go wrong?
#2
 Katuarial SOA Join Date: Aug 2016 Posts: 19 Over the interval cos(x) is 2 to 1, so you have to multiply the pdf by 2 which will then give you the correct answer.
__________________ Last edited by Katuarial; 09-20-2016 at 11:26 AM..
#3
 toesockshoe Member Non-Actuary Join Date: Dec 2014 Posts: 244 Quote:
 Originally Posted by jchaney $f(x) = \frac{1}{2\pi}$ $y = cos x$ find pdf of y. 1. arccos y = x 2. d/dx arcos y = - 1/((1-x^2)^.5) 3. (1/2pi)*| - 1/((1-x^2)^.5)| = 1/(2pi*(1-x^2)^.5) But, solution is 1/(pi*(1-x^2)^.5) Where did I go wrong?
Note that your method only works perfectly if the function is one to one.... Note cos(x) is not one to one over the interval [0,2pi] so you need to multiple by a constant.....To figure out the constant, it might be more intuitive to solve pdf transformations of non one-to-one functions using the CDF method.... drawing a picture might help.... this problem however is a common transformation and you can memorize to multiple it by 2 over the interval.
#4
 jchaney Member CAS Join Date: Oct 2014 College: Univ. of Washington Posts: 96 Thank you for the response. I think I understand what you are explaining. Because the inverse of f(x) range and domain do not match with a 1 to 1, but rather 1 to 2, then the solution is doubled. Aside from trigonometric identities and square transformation, ( Y=X^2), what other problems will have non 1 to 1 relations?

Now I understand why I was missing so many of these problems. Looks like the range of the original function will also determine where to consider the negative value of the square in the inverse.

I spend days hitting my head against a wall trying to figure out this stuff. -- There is new crack in the wall today.

Last edited by jchaney; 09-21-2016 at 03:09 PM..
#5
 toesockshoe1 Member Non-Actuary Join Date: Dec 2014 Posts: 96 Quote:
 Originally Posted by jchaney Thank you for the response. I think I understand what you are explaining. Because the inverse of f(x) range and domain do not match with a 1 to 1, but rather 1 to 2, then the solution is doubled. Aside from trigonometric identities and square transformation, ( Y=X^2), what other problems will have non 1 to 1 relations? Now I understand why I was missing so many of these problems. Looks like the range of the original function will also determine where to consider the negative value of the square in the inverse. I spend days hitting my head against a wall trying to figure out this stuff. -- There is new crack in the wall today.
trig functions and y=x^2 were the only ones ive seen in my practice..

 Thread Tools Search this Thread Show Printable Version Email this Page Search this Thread: Advanced Search Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Posting Rules You may not post new threads You may not post replies You may not post attachments You may not edit your posts BB code is On Smilies are On [IMG] code is On HTML code is Off

All times are GMT -4. The time now is 08:41 AM.

 -- Default Style - Fluid Width ---- Default Style - Fixed Width ---- Old Default Style ---- Easy on the eyes ---- Smooth Darkness ---- Chestnut ---- Apple-ish Style ---- If Apples were blue ---- If Apples were green ---- If Apples were purple ---- Halloween 2007 ---- B&W ---- Halloween ---- AO Christmas Theme ---- Turkey Day Theme ---- AO 2007 beta ---- 4th Of July Contact Us - Actuarial Outpost - Archive - Privacy Statement - Top