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Math Help - proving or disproving probabillity density function

  1. #1
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    proving or disproving probabillity density function

    Prove or disprove: If f1(x) and f2(x) are probability density functions and if θ1 + θ2 = 1, then θ1f1(x) + θ2f2(x) is a probability density function.


    I do not know how to do this at all. Any help would be greatly appreciated.
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  2. #2
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    Quote Originally Posted by cielo View Post
    Prove or disprove: If f1(x) and f2(x) are probability density functions and if θ1 + θ2 = 1, then θ1f1(x) + θ2f2(x) is a probability density function.


    I do not know how to do this at all. Any help would be greatly appreciated.
    1. \int \theta_1 f_1(x) \, dx + \int \theta_2 f_2(x) \, dx = 1 where the integrals are over the supports of each pdf.

    But ....

    2. Is \theta_1 f_1(x) + \theta_2 f_2(x) \geq 0 where \theta_1 + \theta_2 = 1 always true ....?
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    How nice! Thank you so much for the help!
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