SECTION 6.2 Continuous Random Variables 361
- number of days between accidents at a given intersection.
Exercises
- Prove the assertions made above concerning the exponential ran-
dom variable X with density f(t) = λe−λt, t ≥ 0, viz., that
E(X) = 1/λand that Var(X) = 1/λ^2. - Suppose that the useful lifeT of a light bulb produced by a partic-
ular company is given by the density functionf(t) = 0. 01 e−^0.^01 t,
wheretis measured in hours. Therefore, the probability that this
light bulb fails somewhere between timest 1 andt 2 is given by the
integralP(t 1 ≤T ≤t 2 ) =
∫t 2
t 1 f(t)dt.
(a) The probability that the bulb will not burn out beforethours
is a function oftand is often referred to as thereliabilityof
the bulb.
(b) For which value oftis the reliability of the bulb equal to 1/2.
Interpret this value oft.- Suppose that your small company had a single secretary and that
she determined that one a given day, the measured time elapsed
between 30 consecutive received telephone calls was (in minutes)
6.8, 0.63, 5.3, 3.8, 3.5, 7.2, 16.0. 5.5, 7.2, 1.1, 1.4, 1.8, 0.28, 1.2,
1.6, 5.4, 5.4, 3.1, 1.3, 3.7, 7.5, 3.0, 0.03, 0.64, 1.5, 6.9, 0.01, 4.7,
1.4, 5.0.
Assuming that this particular day was a typical day, use these data
to estimate the mean wait time between phone calls. Assuming
that the incoming phone calls roughly follow an exponential dis-
tribution with your estimated mean, compute the probability that
after a given call your secretary will receive another call within
two minutes. - Under the same assumptions as in the above exercise, roughly how
long will it take for the tenth call during the day will be taken by
your secretary?