Determining k of probability density function?
Consider the following probability density function for wind speed:
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
- schmisoLv 71 decade agoFavorite Answer
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
∫ k∙(V/V_max) dV = 1
[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