Sponsored Ads
D.W. Simpson
Worldwide Actuarial Recruitment
Life Health Pension Casualty US Asia Australia Europe Salary Apply
Ezra Penland Actuarial Recruiters
Top Actuary Jobs
Salary Surveys Apply Bios Casualty Health Life Pension
Pauline Reimer, ASA, MAAA
Pryor Associates
Actuarial Openings: Life, P&C, Health, Pensions, Finance
Advertise Here
Contact us at actuary@actuary.com or 770-425-8576
Reach top actuarial professionals

Results 1 to 3 of 3

Thread: Transformation question

  1. #1
    Actuary.com - Newbie Poster
    Join Date
    Jul 2013
    Location
    South Carolina
    Posts
    2

    Transformation question

    I would appreciate if anyone could help with this question. It is from Probability and Statistics with Applications: A Problem Solving Text Asimow and Mawell
    8-3
    Suppose that the random variable X has an exponential distribution with mean one. Let the random variable Y= sqrt (X)

    (a) Find the probability density function for Y.
    (b) Find the expected value for Y.

    For part (a) I got the density function for Y: f(y)= (density function for X)*(dx/dy) = e^-x * 2y= 2ye^(-y^2)

    Using that answer I attempted to solve part (b) by doing the integral from 0 to infinity of y*f(y). But I cannot figure out a solution using integration tables or integration by parts. I suspect I've made an error in the density function of Y.

  2. #2
    Actuary.com - Posting Master
    Join Date
    Oct 2007
    Posts
    3,110
    Quote Originally Posted by bhopkins View Post
    I would appreciate if anyone could help with this question. It is from Probability and Statistics with Applications: A Problem Solving Text Asimow and Mawell
    8-3
    Suppose that the random variable X has an exponential distribution with mean one. Let the random variable Y= sqrt (X)

    (a) Find the probability density function for Y.
    (b) Find the expected value for Y.

    For part (a) I got the density function for Y: f(y)= (density function for X)*(dx/dy) = e^-x * 2y= 2ye^(-y^2)

    Using that answer I attempted to solve part (b) by doing the integral from 0 to infinity of y*f(y). But I cannot figure out a solution using integration tables or integration by parts. I suspect I've made an error in the density function of Y.
    Is the answer \sqrt{\pi}/4? If so you might have to get it in form of the standard normal and then see which factor is missing and use the symmetry of standard normal random variable.

  3. #3
    Actuary.com - Newbie Poster
    Join Date
    Jul 2013
    Location
    South Carolina
    Posts
    2
    Thank you for replying. I finally figured it out sorry I didn't update the thread.

    the answer is sqrt(pi)/2 I had to use an improper integral that converges I found in an earlier chapter it was the integral from negative infinity to positive infinity or e^(-(z/2)^2) is equal to sqrt(2 pi). Or half of that since the integral starts at zero. I integrated sqrt(x)*f(x) instead of the transformation y*f(y) and did a u-substitution. . . then integration by parts.

    As it turns out after reading more in that previous chapter, there is a property of gamma distribution, The integral from zero to infinity or x^(a-1)*e^(-(1/b)x) is equal to b^(a) * gamma(a) . which would have be a lot quicker.
    There was another integral I came across that would have helped me if I tried to integrate y*f(y) but I can't find it. I will go back and look at your way just to see, but the key was finding a improper integral with e in it that equaled some thing with pi in it. I was looking at a calculus book for the longest but i guess it was a probability specific problem.

    I very much appreciate your help. It seems you have no more exams, so I'll try to pass the kindness on by help someone else if I am able.

Thread Information

Users Browsing this Thread

There are currently 1 users browsing this thread. (0 members and 1 guests)

Similar Threads

  1. Variable transformation question
    By bubbarobert8 in forum SOA Exam P / CAS Exam 1 - Probability - with practice exam problems
    Replies: 10
    Last Post: February 7th 2012, 11:22 AM
  2. Transformation of RV Question
    By Tanu in forum SOA Exam P / CAS Exam 1 - Probability - with practice exam problems
    Replies: 4
    Last Post: January 30th 2011, 06:03 PM
  3. PDF transformation problem
    By chicoG in forum SOA Exam P / CAS Exam 1 - Probability - with practice exam problems
    Replies: 3
    Last Post: October 26th 2010, 12:14 AM
  4. SOA 127 # 108 using multivariate transformation?
    By Jo_M. in forum SOA Exam P / CAS Exam 1 - Probability - with practice exam problems
    Replies: 14
    Last Post: July 10th 2008, 05:45 PM
  5. Transformation problem
    By JGET in forum SOA Exam P / CAS Exam 1 - Probability - with practice exam problems
    Replies: 3
    Last Post: August 5th 2007, 11:38 PM

Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •