Distributions with simple truncated expectations

Question

For a project I'm looking for continuous distributions which have a somewhat simple closed form for upper-truncation expectation ($E[x|x>c]$).

Here are two examples I've found so far:

• Exponential distribution $\left(F(x)=1-e^{-\lambda x}\right)$: $\ c+1/\lambda$

• Uniform distribution on $(a,b)$: $\ \frac{c+b}{2}$

Answer 1

Assuming positive $c$:

The logistic distribution with mean 0, scale parameter $s$ has truncated expectation $$-ck + s(1+k)\log(1+k),\text{ where }k=e^{c/s}$$

The Laplace distribution with mean 0, scale parameter $b$ has truncated expectation $$(b+c)(1+e^{-c/b})/2$$

Answer 2

The class of distributions with this property is large (and not even completely defined - does an answer in terms of the gamma function, error function etc. count as closed form?). But note that

$$E(X|X > c) = \frac{\int_c^\infty x f(x) dx}{\int_c^\infty f(x) dx} $$

Therefore a closed form for $E(X|X>c)$ will exist so long as both integrals can be found in closed form.