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Math Help - probability density function

  1. #1
    Junior Member
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    probability density function

    Hey,

    Can anyone help me with the following question:

    f(x) = K(1+x)^n such that x>0
    and
    f(x) = 0 otherwise.

    for some constants K and n.

    For what values of n can f be made into a valid probability density?

    Any help would be greatly appreciated.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by jedoob
    Hey,

    Can anyone help me with the following question:

    f(x) = K(1+x)^n such that x>0
    and
    f(x) = 0 otherwise.

    for some constants K and n.

    For what values of n can f be made into a valid probability density?

    Any help would be greatly appreciated.
    A function is defined as follows:

    <br />
f(x)=\left{ \begin{array}{cc} K(1+x)^n & \mbox{,\ if}\ x>0\\<br />
0 & \mbox{,\ if}\ x \le 0 \end{array}\right<br />
,

    for some K and n.

    For what values of n can f be made into a valid probability density.

    Now f is a probability density if and only if f(x)\ge\ 0 for all x, and

    <br />
\int_{-\infty}^{\infty}f(x) dx = 1<br />
.

    Now the integral is infinite when n \ge \ 0, and if n < \ 0:

    <br />
\int_{-\infty}^{\infty}f(x) dx = \frac{-K}{n+1}=1<br />
.

    Now K > \ 0 as otherwise f(x) would not be be
    greater than or equal to zero. Then n+1=-K, which can only
    be made to work if n <0.

    RonL
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  3. #3
    Junior Member
    Joined
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    Thanks

    Thanks a million.
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