# Distribution of life time of a serial circuit with bulbs

Assume that we have a serial circuit with three bulbs. Each bulb's life time is exponentially distributed: f_{bulb}(t) =\left\{ \begin{aligned} &\lambda e^{-\lambda t} & t \ge 0\\ &0 & t < 0 \end{aligned} \right. Find the distribution function of the circuit's work-time (the circuit works when all three bulbs work). \begin{align} t &= \min \{t_1,t_2,t_3\}\\ F_{circuit}(t) &= P(T_1

I'm stuck here. Did I do this correctly? If yes how do i proceed?

(I don't know how to process the $$\min\{t_1,t_2,t_3\}$$ part)

Sorry if i have bad grammar or messy notation.

• Hint: If you have three independent events $A$, $B$, $C$, then $P(A$ and $B$ and $C)=P(A)\cdot P(B) \cdot P(C)$. Let $A$ be the event that $T_1<t$, let $B$ be the event that $T_2<t$, and let $C$ be the event that $T_3<t$. – irchans Apr 18 at 8:52
• Sorry, that wasn't the part that bugging me, i edited the problem to be more specific. – Hùng Nguyễn Apr 18 at 9:00
• This is relevant: math.stackexchange.com/questions/580279/… – saulspatz Apr 18 at 9:07
• @saulspatz That was extremely helpful. Thank you. – Hùng Nguyễn Apr 18 at 9:38