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Math Help - Maximizing the probability with standard normal random variables

  1. #1
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    Maximizing the probability with standard normal random variables

    Let Z be a standard normal random variable and \alpha be a given constant. Find the real number x that maximizes P( x < Z < x+\alpha).

    Not sure where to even begin other than just looking at the probability function for a normal random variable. Any help or explanation is much appreciated.
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  2. #2
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    Quote Originally Posted by Zennie View Post
    Let Z be a standard normal random variable and \alpha be a given constant. Find the real number x that maximizes P( x < Z < x+\alpha).
    I assume that Z has \mu=0~\&~\sigma=1 and \alpha>0.
    If that is the case, we want to choose x so that 0 is the midpoint of the interval [x,x+\alpha].

    To see why, look at the graph.
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  3. #3
    MHF Contributor matheagle's Avatar
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    Use the fundamental theorem of calculus and differentiate the integral of the normal density over that interval.

    That leads to  e^{-(x+\alpha)^2/2}= e^{-x^2/2} making x=-\alpha/2

    which makes sense, since this straddles the origin.
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