Final answer:
The CDF of the Gumbel distribution can be found by finding the CDF of the exponential distribution and transforming it.
Step-by-step explanation:
The Gumbel distribution is the distribution of -log X, where X follows the exponential distribution with a decay parameter of 1. The cumulative distribution function (CDF) of the Gumbel distribution can be obtained by finding the CDF of the exponential distribution and transforming it.
Let's start with X ~ Exp(1), where the CDF is given by P(X ≤ x) = 1 - e^(-x). To find the CDF of the Gumbel distribution, we substitute -log X in place of x, giving P(-log X ≤ x) = P(X ≥ e^(-x)).
Since the exponential distribution is continuous and monotonically decreasing, we can rewrite this as P(X ≥ e^(-x)) = 1 - P(X < e^(-x)). Using the CDF of the exponential distribution, we have P(X ≥ e^(-x)) = 1 - (1 - e^(-e^(-x))). Therefore, the CDF of the Gumbel distribution is 1 - (1 - e^(-e^(-x))).