The Cauchy distribution with parameters defined by the mode=θ and inter-quartile range, IQR=λ, has pdf given by:
and cdf given by:
If we compare this expression to the t-distribution with 1 degree of freedom (DF), we see that the two functions are the same if θ=0 and λ=1 (a standard Cauchy):
Since the t-distribution is defined by a ratio and has only 1 DF in this case, it can be seen that the Cauchy can be characterized as the ratio of two unit Normal distributions, U/V, and that the ratio V/U is also a Cauchy distribution. However, the Cauchy can arise as the ratio of other identical distributions and in connection with certain problems in geometrical probability (see further, Johnson and Kotz, 1970, ch.16, and the Mathworld entry for the Cauchy distribution). The distribution is symmetric, similar to the Normal but with heavier tails — as an example, for a standard Cauchy distribution the 97.5% point is at 12.7 as compared with 1.96 for a unit Normal.
The Cauchy distribution is sometimes described as pathological, because it has no mean value, no variance and no higher moments or moment generating function. Although this sounds counter-intuitive it follows from the definition of the moments (the expected value integrals for a Cauchy distribution are all infinite), and from the observation that the mean of a sample from a Cauchy distribution has the same distribution as any of its contributing values, so there is no additional information provided by the sample mean. Hence, for example, a random sample from a Cauchy distribution with mode at 0 may well yield a mean value of 3, 5 or -3, depending on the sample taken. The mode and median, which are equal, are useful however, as is the IQR. In the graph below we plot the standard Cauchy against the Normal distribution with the same mode and matching upper and lower quartiles at +/-1.
Standard Cauchy distribution compared with matching Normal
Mathworld: Weisstein E W: Cauchy distribution: https://mathworld.wolfram.com/CauchyDistribution.html
Wikipedia: Cauchy distribution: https://en.wikipedia.org/wiki/Cauchy_distribution