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Math Help - Determining the CDF of the Laplace PDF

  1. #1
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    Determining the CDF of the Laplace PDF

    \frac{1}{2} \int^x_{- \infty} e^{-|t|} dt= \frac{1}{2} \int^x_{ \infty} e^{t}dt + \frac{1}{2} \int^x_{- \infty} e^{-t} dt

    I'm pretty sure I bounded the second integral incorrectly. What would be the correct way?
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  2. #2
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    Hello,

    First you have to determine if x is positive or negative...
    Let's suppose it's negative.
    Then t\in(-\infty,x)\implies |t|=-t

    So \int_{-\infty}^x e^{-|t|} ~dt=\int_{-\infty}^x e^t ~dt=\int_{-x}^\infty e^{-t} ~dt (if you substitute t=-t)

    If x is positive, then t\in(-\infty,0) \implies |t|=-t and t\in(0,x)\implies |t|=t

    So \int_{-\infty}^x e^{-|t|} ~dt=\int_{-\infty}^0 e^{t} ~dt+\int_0^x e^{-t} ~dt=1+\int_0^x e^{-t} ~dt
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  3. #3
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    Thank you once again.
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