# Distributions with simple truncated expectations

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 ($$F(x)=1-e^{-\lambda x}$$): $$c+1/\lambda$$

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

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 $$(b+c)(1+e^{-c/b})/2$$

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.

• That makes sense. I guess I was looking for simplest examples since the expectation fits into a differential equation and that is ultimately what I hope to have a closed form solution of. – user180743 Oct 10 at 11:58