1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Stats/prob: finding cumulative distribution function

  1. Feb 27, 2013 #1
    1. The problem statement, all variables and given/known data

    given pdf:

    f(x) = 2/3x for 0<=x<=1
    f(x) = 2/3 for 1<x<=2
    f(x) = 0 elsewhere

    Find the CDF.

    2. Relevant equations

    3. The attempt at a solution

    I've found:

    F(x) = 0 for x<= 0
    F(x) = 1 for x>=2
    F(x) = (1/3)x2 for 0<=x<=1

    and I found:

    F(x) = (2/3)x for 1<x<=2

    However, the last bit is incorrect. It should be F(x) = (2/3)x -(1/3)

    I'm unclear as to why. I think it has something to do with solving for the constant of integration, but I'm not sure exactly.
  2. jcsd
  3. Feb 27, 2013 #2


    User Avatar
    Homework Helper

    Yes, it has to do with the integration constant. Your CDF has to be continuous, so you need to fix that constant value so that

    $$\lim_{\epsilon \rightarrow 0} F(1 - \epsilon) = \lim_{\epsilon \rightarrow 0} F(1 + \epsilon).$$

    That is, the limit of the CDF on either side of x = 1 have to be the same, which wouldn't be true if you didn't fix the constant of integration to be -1/3 just above x=1.
  4. Feb 27, 2013 #3
    Could you clarify.. what is epsilon?
  5. Feb 27, 2013 #4

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    You have F correct for x ≤ 1 and for x ≥ 2. Since the random variable has a finite density function, its F(x) must be a continuous function, and since F'(x) = 2/3 on [1,2], F must increase linearly with slope 2/3, starting from F(1) = 1/3 and ending at F(2) = 1. You can figure out what the formula must be for F(x) in the region 1 ≤ x ≤ 2.

    Basically, you need to use
    [tex] F(x) = F(1) + \int_{1}^{x} f(t) \, dt, \: 1 \leq x \leq 2.[/tex]
    Note that in this calculation there is NO constant of integration!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted