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Cauchy Distribution

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Find the probability density function, lower cumulative distribution function, and upper cumulative distribution function of the Cauchy distribution.
Percentile x
Location parameter a
Scale parameter b
b＞0
\(\normalsize Cauchy\ distribution\\ (1)\ probability\ density\\ \hspace{30px}f(x,a,b)={\large\frac{1}{\pi}\frac{b}{(x-a)^2+b^2}}\\ (2)\ lower\ cumulative\ distribution\\ \hspace{30px}P(x,a,b)={\large\int_{\small-\infty}^{\small x}}f(t,a,b)dt\\ (3)\ upper\ cumulative\ distribution\\ \hspace{30px}Q(x,a,b)={\large\int_{\small x}^{\small\infty}}f(t,a,b)dt\\\)

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