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Distribution function

  1. Jan 29, 2009 #1
    X is continuously distributed with probability density

    [tex]f_{X}(x) = nx^{n-1}, if 0 < x \leq 1[/tex]
    and
    [tex]f_{X}(x) = 0, otherwise[/tex]

    Find the distribution function F(x) of X. Find the probability that X lies between 0.25 and 0.75 when n=1 and when n=2. Find the median of X, i.e. the value of a so that [tex]P(X \leq a) = 1/2[/tex], when n=1 and when n=2. Find E(X) when n=1 and when n=2 and compare with the corresponding medians.


    I'm first gonna try to find the distribution function. Does this simply mean finding n in the probability density?
     
  2. jcsd
  3. Jan 29, 2009 #2

    statdad

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    Homework Helper

    If [tex] f(x) [/tex] is the density function for a random variable, the distribution function is

    [tex]
    F(x) = \int_{-\infty}^x f(t) \, dt
    [/tex]

    You can calculate [tex] P(a \le X \le b) [/tex] either by

    [tex]
    F(b) - F(a)
    [/tex]

    or

    [tex]
    \int_a^b f(x) \, dx
    [/tex]

    (these are actually the same things dressed up in different notations)

    To find the median solve [tex] F(a) = 0.5 [/tex]

    To find the expected values calculate integrate [tex] x f(x) [/tex].
     
  4. Jan 29, 2009 #3
    [tex]
    F(x) = \int_{-\infty}^x nt^{n-1} \, dt = [t^n]^{x}_{- \infty}
    [/tex]

    Shoult the integral limits be from 0 to x instead of negative infinity to x? I don't know, but that's the way it's supposed to be done if I interppret an example in my book correctly. Then I get

    [tex]
    F(x) = x^n
    [/tex]

    and

    P(0.25 < X < 0.75) = 0.5 for both n=1 and n=2.

    and the medians a=0.5 and a=0.707 when n=1 and n=2, respectively

    and finally

    E(X) = 0.5 and E(X) = 2/3 when n=1 and n=2, respectively.
     
    Last edited: Jan 29, 2009
  5. Jan 30, 2009 #4

    statdad

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    Homework Helper

    Yes - I gave the general definitions for your needs. Since your density is zero outside of the interval [tex] [0,1] [/tex], every

    [tex]
    \int_{-\infty}^\infty \text{ stuff} \, dx
    [/tex]

    reduces to

    [tex]
    \int_0^1 \text{stuff} \, dx
    [/tex]
     
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