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Math Help - iid uniform

  1. #1
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    iid uniform

    Assume that X_1,X_2,...,X_n are iid uniform on [0,1]. Show that (X_1X_2...X_n)^\frac{1}{n}\to e^{-1} a.e. for n\to\infty.

    I appreciate any help.
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  2. #2
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    Quote Originally Posted by Veve View Post
    Assume that X_1,X_2,...,X_n are iid uniform on [0,1]. Show that (X_1X_2...X_n)^\frac{1}{n}\to e^{-1} a.e. for n\to\infty.

    I appreciate any help.
    Express the power using a^b=e^{b\log a} and use the law of large numbers.

    (Or equivalently take the logarithm of the sequence and apply the law of large numbers)
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  3. #3
    MHF Contributor matheagle's Avatar
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    you also need to show that

    E(\ln X)=\int_0^1 \ln xdx=-1
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  4. #4
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    Thanks. That I was able to prove.
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