Exercise 5.33 Find the distribution function of the so-called ‘extreme value’ density function f (x) = exp(−x − e−x ) for x ∈ R.

5.4 Some common density functions It is fairly clear that any function f which satisfies f (x) ≥ 0 for x ∈ R (5.34)

and Z

∞

−∞

f (x) dx = 1 (5.35)

is the density function of some random variable. To confirm this, simply define F(x) =

Z x

−∞

f (u) du and check that F is a distribution function by verifying (5.5)–(5.8). There are several such functions f which are especially important in practice, and we list these below.

The uniform distribution on the interval

(a,

b) has density function f (x) =





1 b − a if a < x < b, 0 otherwise.

(5.36)