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Show f is a probability density function

  1. Jan 17, 2010 #1
    1. The problem statement, all variables and given/known data

    Let f(x) = (1 + ux)/2 for -1<= x <= 1 and 0 otherwise . where -1<= u <= 1

    a) show f is a density


    2. Relevant equations

    TO show
    1. f(x) >= 0

    2. intergeral f (from -infinity to infinity) = 1

    3. The attempt at a solution

    I have done 2. and proved that it is 1 by taking the intergeral of f.

    However, how do u show f >= 0.

    i have found the dervavtive of f to be u / 2 for -1<= x <= 1 and 0 otherwise.

    I dun know where to go?
     
  2. jcsd
  3. Jan 17, 2010 #2

    Dick

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    If you've shown f'(x)=u/2 then that means the max and min of f(x) must be at x=(-1) or x=1, the boundaries of your interval. Can you take it from there?
     
  4. Jan 17, 2010 #3
    is this also possible?

    1<=x<=1
    -1<=ux<=1
    0<=1+ux<=2
    0<=(1+ux)/2<=1
    therefore its always >= 0
     
  5. Jan 18, 2010 #4

    Dick

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    That works too.
     
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