Joint Density

Discussion in 'Probability' started by Tims, Oct 2, 2010.

  1. Tims

    Tims Guest

    Let X1,X2,X3, and X4 be independent continuous random variables with
    a common distribution function F and let
    p = P{X1 <X2 >X3 <X4}

    (a)
    Find p by integrating the joint density function over the appropriate
    region.

    (b)
    Find p by using the fact that all 4! possible orderings of X1, . . . ,
    X4 are equally likely.

    [Difficulty]
    I have starting problem with (a)

    (b) seems to be easy

    [Thoughts]
    For (b),

    X1 __ X2 __ X3 __ X4

    The symbols can be arranged in 4! ways.
    The signs in each blank can be either <, = or >=

    The blanks need to be filled with <, > and < respectively,
    SO, P(<) = P(>) = P(=) = 1/3

    So (b) must be (1/3)^3 times 4!
     
    Tims, Oct 2, 2010
    #1
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