📐 Cauchy Distribution Graph 計算機

The Cauchy distribution, also called the Lorentz distribution, is a continuous probability distribution defined by a location parameter a (the median and peak position) and a scale parameter b > 0 (the half-width at half-maximum). It is famous for its heavy tails: it has no finite mean and no finite variance. This calculator evaluates the distribution over a sequence of x values so you can build a plot-ready table of (x, value) pairs.

Pick a function: probability density f, lower cumulative probability P, or upper cumulative probability Q. Enter the location a and scale b (must be positive). Then define the x sequence with an initial value, a step increment, and the number of points. The k-th x is x_k = xInitial + k * xStep for k = 0 to count-1. The defaults sweep x from -5 to +5 in steps of 0.1 (101 points).

Let z = (x - a) / b. The density is

equivalently ((1) ÷ (π b (1 + z^2))). The lower cumulative distribution is and the upper (survival) function is Because arctan stays within (-π/2, π/2), both P and Q remain strictly between 0 and 1.

With a = 0, b = 0.7, evaluate density at x = 1: (x-a)^2 + b^2 = 1 + 0.49 = 1.49, so

For the lower cumulative at the same point, \arctan(1/0.7) = 0.96007, so and Q = 1 - 0.80559 = 0.19441. At the peak x = a, f = ((1) ÷ (π b)) and P = Q = 0.5.