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Determining k of probability density function?

Consider the following probability density function for wind speed:

http://www.imagedump.com/index.cgi?pick=get&tp=549...

What is an appropriate value of k for this to be a legitimate probability density function?

note: I know f(V) is supposed to be (k/c)*(V/c)^k-1*e^(-(V/c)^k) but can't figure out how to obtain k

1 Answer

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  • 1 decade ago
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    The formula:

    f(V) = (k/c)∙(V/c)^(k-1)∙e^(-(V/c)^k)

    represents a Weibull distribution, commonly used for wind distribution.

    The given distribution is apparently not a Weibull distribution. Just compare your graph with the WD graphs shown in the wikipedia article linked below.

    What you have is a linear distribution, which can defined be written as follows:

    f(V) = k∙(V/V_max) for 0≤V≤V_max

    f(V) = 0 x>V_max

    with V_max = 10m/s

    Because of the probabilities of a all possible outcomes of an event sum up to unity, any probability distribution f(x) of a variable x has to satisfy the condition:

    ∫ f(x) dx = 1

    -∞

    Plug in your wind speed distribution and solve the integral. Then you get an equation for k, such that f(V) satisfies condition above:

    ∫ f(V) dV = 1

    -∞

    restrict to integral to th range where f(V) is not zero

    <=>

    V_max

    ∫ k∙(V/V_max) dV = 1

    0

    <=>

    [k/(2∙V_max)] ∙(V_max)² - [k∙/(2∙V_max)] ∙(0)² = 1

    <=>

    (k/2)∙V_max = 1

    <=>

    k = 2/V_max = 2 / 10m/s = 0.2s/m

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