So I'm given this function:
$$ f(t) = 0.5e^{-0.5t} , t > 0 $$
I'm trying to find mu:
$$ \mu = \int_0^{\infty}tf(t)dt $$
Integrating by parts I end up with:
$$ \mu = -e^{-0.5t}(t+2)\bigg|_{0}^{\infty} $$
I'm starting to think, while I write this, that I should take the limit of big N as N goes to infinity to find my mu, but I'm kinda at a loss here.
Consider $F(t) = -e^{-.5 t}(t + 2)$; then by L'Hospital's rule,
$$\lim_{t \to \infty} F(t) = \lim_{t \to \infty} -\frac{t + 2}{e^{.5 t}} = \lim_{t \to \infty} -\frac{1}{.5 e^{.5 t}} = 0$$
Hence, $$\mu = \lim_{t \to \infty} \int_0^{\infty} tf(t) dt = \lim_{t \to \infty} F(t) - F(0) = 0 + e^{0}(0 + 2) = 2$$