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Thread: Maximizing the probability with standard normal random variables

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

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

    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 $\displaystyle Z$ be a standard normal random variable and $\displaystyle \alpha$ be a given constant. Find the real number $\displaystyle x$ that maximizes $\displaystyle P( x < Z < x+\alpha)$.
    I assume that $\displaystyle Z$ has $\displaystyle \mu=0~\&~\sigma=1$ and $\displaystyle \alpha>0$.
    If that is the case, we want to choose $\displaystyle x$ so that $\displaystyle 0$ is the midpoint of the interval $\displaystyle [x,x+\alpha]$.

    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 $\displaystyle e^{-(x+\alpha)^2/2}= e^{-x^2/2}$ making $\displaystyle x=-\alpha/2$

    which makes sense, since this straddles the origin.
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