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Math Help - Laplace distribution

  1. #1
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    Laplace distribution

    The Laplace distribution, also known as a double exponential, has a pdf given by:

    Find the theoretical mean and variance of a laplace distribution. (Hint: Integrals of absolute values should be done as a positive and negative part, in this case, with limits from -∞ to μ and from μ to ∞.)
    I am not sure how to even go about this problem. I think you would have to set it as an integral using the following formula for the expected value of the mean: (lambda would be a constant)



    If this is correct how would you solve this integral?

    Thanks
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  2. #2
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    Re: Laplace distribution

    Quote Originally Posted by acasas4 View Post
    The Laplace distribution, also known as a double exponential, has a pdf given by:

    Find the theoretical mean and variance of a laplace distribution. (Hint: Integrals of absolute values should be done as a positive and negative part, in this case, with limits from -∞ to μ and from μ to ∞.)
    I am not sure how to even go about this problem. I think you would have to set it as an integral using the following formula for the expected value of the mean: (lambda would be a constant)



    If this is correct how would you solve this integral?

    Thanks
    You're expected to know, and the hint explicitly directs you to this, that

    f(x) = \frac{\lambda}{2} e^{-\lambda (x - \mu)} if x - \mu > 0 \Rightarrow x > \mu

    f(x) = \frac{\lambda}{2} e^{-\lambda (\mu - x)} = \frac{\lambda}{2} e^{\lambda (x - \mu)} if x - \mu < 0 \Rightarrow x < \mu

    and break up the integral accordingly.
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