A.20 Fisher’s z distribution

X has a z distribution on and degrees of freedom, denoted

if , or equivalently if it has the density

Another definition is that if , then

The mean and variance are easily deduced from those of the log chi-squared distribution; they are

where is (as above) the digamma function, or approximately

The mode is zero.

Unless and are very small, the distribution of z is approximately normal. The z distribution was introduced by Fisher (1924).