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Math Help - Density Functions

  1. #1
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    Density Functions

    Q: Consider the function f(x) = (e^-|x|)/2 for x belonging to Real numbers. Prove that f is a density function.

    Approach:
    For f(x) to be classified as a density function there are 2 things required:
    1. f(x) >= 0 for all x belonging to Real numbers.
    2. Indefinite Integral f(x)dx = 1.

    Property 1 is simple enough to prove as f(x) >= 0 for all values of x. However, I am stumped on proving property 2. As far as I can tell, the integral is undefined. Am I going about the solution of this question all wrong or is there something else that proves a density function?
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    MHF Contributor harish21's Avatar
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    For functions involving absolute values, you can break them up into two different functions and then work through as two separate cases.

    So, if you are to integrate the function you have written, then:

    For x > 0, you can replace |x| by x , that is, integrate (e^-x)/2 dx.

    For x < 0, you can replace |x|by -x, that is, integrate (e^x)/2 dx
    Last edited by harish21; October 7th 2010 at 01:23 PM.
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    \displaystyle \int_{ - \infty }^\infty  {\frac{{e^{ - |x|} }}<br />
{2}dx}  = 1

    Notice that f(x) = \dfrac{{e^{ - |x|} }}{2} is an even function.
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  4. #4
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    Quote Originally Posted by ipacheco View Post
    Q: Consider the function f(x) = (e^-|x|)/2 for x belonging to Real numbers. Prove that f is a density function.

    Approach:
    For f(x) to be classified as a density function there are 2 things required:
    1. f(x) >= 0 for all x belonging to Real numbers.
    2. Indefinite Integral f(x)dx = 1.

    Property 1 is simple enough to prove as f(x) >= 0 for all values of x. However, I am stumped on proving property 2. As far as I can tell, the integral is undefined. Am I going about the solution of this question all wrong or is there something else that proves a density function?
    Thread of related interest: http://www.mathhelpforum.com/math-he...tml#post566813
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  5. #5
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    Thanks everyone. I used the two case method where I split up the integral based on whether x was positive or negative. It became simple enough after that. Thanks for the help.
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