# 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

- schmisoLv 71 decade agoFavorite Answer
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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