Results 1 to 6 of 6

Math Help - Probability Generating Functions

  1. #1
    Newbie
    Joined
    Jun 2008
    Posts
    24

    Probability Generating Functions

    Can I please have some help on this subject?

    I have a continous random variable given by

    f(x)=k(3/8)^x

    and have been asked to find the probabilty generating function Gx(N) and so far have:

    Gx(N)=SUM(0 -> infinity) k(n^x)(3/8)^x
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by SilenceInShadows View Post
    Can I please have some help on this subject?

    I have a continous random variable given by Mr F says: Surely you mean discrete random variable ......

    f(x)=k(3/8)^x

    and have been asked to find the probabilty generating function Gx(N) and so far have:

    Gx(N)=SUM(0 -> infinity) k(n^x)(3/8)^x
    G_x(n) = k \sum_{x=0}^{\infty} n^x \left( \frac{3}{8} \right)^x = k \sum_{x=0}^{\infty} \left( \frac{3n}{8} \right)^x = \frac{k}{1 - \frac{3n}{8}} = \frac{8k}{8 - 3n}

    since the sum is that for an infinite geometric series (and is not finite unless 0 < \frac{3n}{8} < 1).

    To get the value of k, you use the infinite geometric series again:

    k \sum_{x=0}^{\infty}\left( \frac{3}{8} \right)^x = 1 \Rightarrow \frac{k}{1 - \frac{3}{8}} = 1 \Rightarrow k = \frac{5}{8}.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jun 2008
    Posts
    24
    Thanks, with a follow up question;
    The wiki page on this is confusing me, to find the expectation;
    do I differentiate and put n=1?
    and if so is it the same approach for variance?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Hello,

    Quote Originally Posted by SilenceInShadows View Post
    Thanks, with a follow up question;
    The wiki page on this is confusing me, to find the expectation, do I differenttiate and put n=1? and if so is it the same approach for Variance?
    If f is the generating function of a random variable X, then :

    f'(1)=\mathbb{E}(X) (like you said)

    and f''(1)=\mathbb{E}(X(X-1))=\mathbb{E}(X^2)-\mathbb{E}(X), by linearity of the expectation.
    This implies : \mathbb{E}(X^2)=f''(1)+\mathbb{E}(X)=f''(1)+f'(1)

    Knowing that \text{var}(X)=\mathbb{E}(X^2)-(\mathbb{E}(X))^2, you can say :
    \text{var}(X)=f''(1)+f'(1)-\left(f'(1)\right)^2

    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Jun 2008
    Posts
    24
    One further thing, its just a clarification on notation
    my question literally says:

    f(x)= xexp(x)

    does that mean f(x) = x*2.718^x ? as in the expotential?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Quote Originally Posted by SilenceInShadows View Post
    One further thing, its just a clarification on notation
    my question literally says:

    f(x)= xexp(x)

    does that mean f(x) = x*2.718^x ? as in the expotential?
    Yes
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Probability Generating Functions
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: August 10th 2009, 02:10 PM
  2. probability generating functions- please help
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: May 11th 2009, 07:01 PM
  3. Probability Generating Functions Question
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: February 15th 2009, 07:39 PM
  4. Probability problem to do with generating functions
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: February 4th 2009, 11:57 AM
  5. Probability and Moment Generating Functions
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: May 22nd 2008, 04:11 AM

Search Tags


/mathhelpforum @mathhelpforum