Lévy distribution
Given a random variable x in the Lévy distribution, find the probability density function f(x), the lower cumulative probability P(x), and the upper cumulative probability Q(x).
| \(\normalsize Lévy\ distribution\ f(x,\mu,c)\\ (1)\ probability\ density\\ \hspace{30px}f(x,\mu,c)= \sqrt{\frac{c}{2\pi}} \exp \left[ {-\frac{c}{2(x-\mu)}} \right] (x-\mu)^{-\frac{3}{2}}\\ (2)\ lower\ cumulative\ distribution\\ \hspace{30px}P(x,\mu,c)={\large\int_{\small 0}^{\small x}}f(t,\mu,c)dt=erfc(\sqrt{c/2(x-\mu)})\\ (3)\ upper\ cumulative\ distribution\\ \hspace{30px}Q(x,\mu,c)={\large\int_{\small x}^{\small\infty}}f(t,\mu,c)dt\\ \) |
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