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In an office building, the lift breaks down randomly at a mean rate of 3 times per week. The random variable X represents the time in days between successive lift breakdowns.

(i) Calculate the probability that the time interval between successive lift breakdowns is between 2 and 3 days.

(ii) Find the probability that, after a breakdown has just occurred, at least 1 week will pass without another breakdown occurring.

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The time $X$ between breakdowns, measured in days, has mean $\frac{7}{3}$, so we are dealing with an exponential with parameter $3/7$, and hence density function $\frac{3}{7}e^{-3x/7}$, for $x\ge 0$. We want the probability that $2\le X\le 3$. This probability is $$\int_{2}^{3} \frac{3}{7}e^{-3x/7}\,dx.$$ Now it is just a matter of integrating.

The second question is in a sense simpler. We want the probability that $X\ge 7$. This is
$$\int_{7}^{\infty} \frac{3}{7}e^{-3x/7}\,dx.$$

Remark: In some courses, for people with little knowledge of calculus, people are just told that if $X$ has exponential distribution with parameter $\lambda$, then $\Pr(X\le x)=F_X(x)=1-e^{-\lambda x}$ (for $x\ge 0$).

In such a course, one would note that the probability that $X$ is between $a$ and $b$, where $0\le a\lt b$ is $F_X(b)-F_X(a)$. This is $(1-e^{-\lambda b})-(1-e^{-\lambda a})$, which simplifies to $e^{-\lambda a}-e^{-\lambda b}$.

Similarly, the probability that $X\gt a$ is $e^{-\lambda a}$.

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  • $\begingroup$ The time X between breakdowns, measured in days, is 7/3... Actually 7/3 is the mean time and the time X itself is random. $\endgroup$ – Did May 25 '13 at 7:46
  • $\begingroup$ @Did: Thanks, verbal slippage. $\endgroup$ – André Nicolas May 25 '13 at 7:48
  • $\begingroup$ @AndréNicolas : sir if you can help me with this question also it will be so much useful in my exam which will held in next week math.stackexchange.com/questions/401810/… $\endgroup$ – ChampR May 25 '13 at 9:29

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