# Help with some practice exam probability questions

I'm working through a practice exam and I'm having a bit of trouble with question 1b. and 2. The exam is at Linky.

1) Suppose that a room is illuminated by two light bulbs, $A$ and $B$. The lifetimes, $L_A$, and $L_B$ of the two bulbs are independent exponential random variables each with mean 2000 hours.

(a) If the bulbs are both switched on at time 0 what is the mean time that the room will stay illuminated if neither bulb is replaced after it burns out. Hint: You need to ﬁnd the density of the random variable $L=\max \{L_A,L_B\}$ and then do an appropriate integral to ﬁnd $\Bbb E(L)$.

(b) Suppose that when either of the bulbs burns out a signal is sent to the maintenance crew that one of the lights burned out. The maintenance crew then schedules the burnt out bulb for replacement within some interval of time $T$. What is the longest this time T can be to still insure that with probability .001 the burnt out bulb is replaced before the second bulb burns out (only 1 in a 1000 times will the room go dark before the bulb is replaced). Hint: Condition on the events $(L_A<L_B)$ and $(L_B <L_A)$.

For 1b. I figured the PDF to be $P(\max(L_A,L_B)<a)=P(L_B<a)P(L_A<a)$ because the two random variables are independent. Because they are exponential, this became $(1-e^{-\lambda a})^2$. Now I took the derivative to find $\rho_{L_A,L_B}(a)=2\lambda e^{-\lambda a}-2\lambda e^{-2\lambda a}$. Then I took the integral from 1 to $\infty$ to find $E(L)=3000$ hours. Is this part correct?

I'm not sure where to begin with 1b. Do I have to use the density function found in #1?

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It is generally preferred for StackExchange questions to be self-contained (i.e not rely on external links). –  Austin Mohr Nov 30 '11 at 2:43
@AustinMohr, sorry. I didn't think I could succinctly describe problem 1b. without the rest of problem 1. –  thezboe Nov 30 '11 at 2:45
The $3000$ looks fine. You don't need to know how to integrate if you know the expression for the mean of exponential distributions, since $\lambda=1/2000$ and $2\lambda=1/1000$. For $1$(b) the easiest way is to exploit the memorylessness of the exponential. When first bulb burns out, second acts like a brand new bulb. (Of course the exponential distribution models lifetime of bulbs not very well. Real bulbs do age.) –  André Nicolas Nov 30 '11 at 7:41
I included the problem statement. –  David Mitra Nov 30 '11 at 8:10