@SiongThyeGoh

while calculating mean total life in page 174

Karlin book

why $E(\delta_{t}) = \int_{0}^{t} (Pr(\delta_{t} > x))dx$ ?

$$E(\delta_t) = \int_0^\infty Pr(\delta_t >x )\, dx = \int_0^t Pr(\delta_t >x )\, dx + \int_t^\infty Pr(\delta_t >x )\, dx=\int_0^t Pr(\delta_t >x )\, dx + 0$$

No why $\delta_{t} < x$ ?

sorry why $\delta_{t} > x$

For nonnegative random variable, this is the formula for expectaiton.

related

Thank you :)

The first answer is so nice and illustrative, this shows me Math is beautiful and I like it and feel like to dance to a bass boosted song :) :)