A.21 Cauchy distribution

X has a Cauchy distribution with location parameter μ and scale parameter , denoted

if

Because the relevant integral is not absolutely convergent, this distribution does not have a finite mean, nor a fortiori, a finite variance. However, it is symmetrical about μ, and hence

The distribution function is

so that when then F(x)=1/4 and when then F(x)=3/4. Thus, and are, respectively, the lower and upper quartiles, and hence σ may be thought of as the semi-interquartile range. Note that for a normal distribution the semi-interquartile range is rather than σ.

It may be noted that the C(0, 1) distribution is also the Student’s t distribution on 1 degree of freedom