対数正規分布

\(\normalsize Lognormal\ distribution\ Log N(x,\mu,\sigma)\\ (1)\ probability\ density\\ \hspace{30px}f(x,\mu,\sigma)= {\large\frac{1}{\sqrt{2\pi}\sigma x}e^{-\frac{1}{2}\left(\frac{\ln(x)-\mu}{\sigma}\right)^2}}\\ (2)\ lower\ cumulative\ distribution\\ \hspace{30px}P(x,\mu,\sigma)={\large\int_{\small 0}^{\small x}}f(t,\mu,\sigma)dt\\ (3)\ upper\ cumulative\ distribution\\ \hspace{30px}Q(x,\mu,\sigma)={\large\int_{\small x}^{\small\infty}}f(t,\mu,\sigma)dt\\ (4)\hspace{20px} mean:\ e^{\mu+\frac{\sigma^2}{2}}\\ \hspace{25px}median:\ e^{\mu}\\ \hspace{40px}mode:\ e^{\mu-\sigma^2}\\ \)