Results 1 to 2 of 2

Math Help - Probability with two random variables

  1. #1
    Newbie
    Joined
    Jun 2009
    Posts
    6

    Probability with two random variables

    Consider two independent random variables: r,s, where
    r\sim f(r) and s \sim U[0,x].
    ( U[0,x] means uniform within support 0 and x.)

    What is the following probability:

    \Pr (r-ts>p), where t and p are constants.

    Any suggestions, tips or ideas?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    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 Justin Lo View Post
    Consider two independent random variables: r,s, where
    r\sim f(r) and s \sim U[0,x].
    ( U[0,x] means uniform within support 0 and x.)

    What is the following probability:

    \Pr (r-ts>p), where t and p are constants.

    Any suggestions, tips or ideas?
    Since they're independent, their joint probability density function is f(r)\cdot \frac 1x

    Now, \mathbb{P}(r-ts>p)=\mathbb{E}\left(\mathbf{1}_{r-ts>p}\right) (1 is the indicator function)

    By the law of the unconscious statistician, which states that for any measurable function f, \mathbb{E}(f(X_1,\dots,X_n))=\int_{\mathbb{R}^n} f(x_1,\dots,x_n) g(x_1,\dots,x_n) ~dx_1\dots dx_n

    where g is the joint probability function of (X_1,\dots,X_n)


    So here, we have :

    \mathbb{P}(r-ts>p)=\int_0^x\int_{\mathbb{R}} \frac{f(r)}{x} \cdot \mathbf{1}_{r-ts>p} ~dr ~ds

    now, \mathbf{1}_{r-ts>p} has to be interpreted as a region.
    Since we first integrate with respect to r, consider it with respect to r :
    r>p-ts

    So finallly :

    \mathbb{P}(r-ts>p)=\int_0^x\int_{p-ts}^\infty \frac{f(r)}{x} ~dr ~ds



    Note : the inner integral doesn't necessarily goes to infinity, it depends on the support of the rv r, which will be expressed in the function f.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Random variables and probability distributions #1
    Posted in the Statistics Forum
    Replies: 0
    Last Post: September 9th 2011, 03:15 AM
  2. Probability of Random Variables
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: October 19th 2010, 06:49 PM
  3. Probability - random variables
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: May 4th 2010, 09:26 PM
  4. Join Probability & Random Variables
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: March 29th 2009, 07:40 PM
  5. probability with random variables
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: May 14th 2008, 08:02 PM

Search Tags


/mathhelpforum @mathhelpforum