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Homework Help: 'Triangular Distributions' Probability Density Function

  1. Oct 9, 2008 #1
    (\Triangular" distributions.) Let X be a continuous random variable with prob-
    ability density function f(x). Suppose that all we know about f is that a </= X </= b,
    f(a) = f(b) = 0, and that there exists a value c between a and b where f is at a maxi-
    mum. A natural density function to consider in this case is a piece-wise linear function,
    corresponding to lines connecting (a; 0) with (c; f(c)), and (c; f(c)) with (b; 0).
    a) What is the value of f(c)?
    b) Sketch a graph of f(x).
    c) Compute the expected value E(X) and the variance Var(X).

    I have not been given any numbers and am very confused as to how there could be a numerical answer to this question. I know the probability density function looks like a triangle, with f(a) and f(b) on the x-axis, but am not sure where to go with this. Anyone have a suggestion?
     
  2. jcsd
  3. Oct 9, 2008 #2

    mathman

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    Science Advisor

    The density function will look like:

    f(x)=K(x-a) a<x<c
    f(x)=L(b-x) c<x<b

    where L and K are determined by:
    L(b-c)=K(c-a)
    integral from a to b of f(x)=1.

    Your results will be functions of a, b and c, so don't expect to get numbers unless a, b, and c are specified.
     
    Last edited: Oct 10, 2008
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