of is of type I, namely . Similarly, if the cumulative distribution function of is of type III, and with the negative numbers as support, i.e. , then the cumulative distribution function of is of type I, namely . Link to logit models (logistic regression) Multinomial logit models, and certain other types of logistic regression, can be phrased as latent variable models with error variables distributed as Gumbel distributions (type I generalized extreme value distributions). This phrasing is common in the theory of discrete choice models, which include logit models, probit models, and various extensions of them, and derives from the fact that the difference of two type-I GEV-distributed variables follows a logistic distribution, of which the logit function is the quantile function. The type-I GEV distribution thus plays the same role in these logit models as the normal distribution does in the corresponding probit models. Properties The cumulative distribution function of the generalized extreme value distribution solves the stability postulate equation. The generalized extreme value distribution is a special case of a max-stable distribution, and is a transformation of a min-stable distribution. Applications The GEV distribution is widely used in the treatment of "tail risks" in fields ranging from insurance to finance. In the latter case, it has been considered as a means of assessing various financial risks via metrics such as value at risk. However, the resulting shape parameters have been found to lie in the range leading to undefined means and variances, which underlines the fact that reliable data analysis is often impossible. In hydrology the GEV distribution is applied to extreme events such as annual maximum one-day rainfalls and river discharges. The blue picture, made with CumFreq, illustrates an example of fitting the GEV distribution to ranked annually maximum one-day rainfalls showing also the 90% confidence belt based on the binomial distribution. The rainfall data are represented by plotting positions as part of the cumulative frequency analysis. Example for Normally distributed variables Let be i.i.d. normally distributed random variables with mean 0 and variance 1. The Fisher–Tippett–Gnedenko theorem tells us that , where . This allow us to estimate e.g. the mean of from the mean of the GEV distribution: where is the Euler–Mascheroni constant. Related distributions If then If (Gumbel distribution) then If (Weibull distribution) then If then (Weibull distribution) If (Exponential distribution) then If and then (see Logistic_distribution). If and then (The sum is