So we are given an infinite supply of bulbs, following Exp($\lambda$)
You arrive at time t, note the bulb that is burning at time t.
$L_t$ = Lifetime of the bulb we noted
$A_t$= Lifetime of the noted bulb spent until time t
$B_t$ = Remaining lifetime of the noted bulb
I was able to calculate the distribution of the $L_t$, which came out as
P[$L_t \leq x$] = $1-((x\lambda + 1)* e^{-\lambda x}$ for $0<x\leq t$ and 1-(($t\lambda + 1)e^{-\lambda x}$ for $x \geq t$
I know $B_t$ follows Exp($\lambda$), which is the paradox.
How to calculate the distribution of $A_t$?
I was trying to use change of variable but because of dependent variable I was unable to proceed