From Algorithms to Z-Scores - matloff - University of California, Davis

9.1. MEMORYLESS PROPERTY OF EXPONENTIAL DISTRIBUTIONS 185
1 − FV (t). Then from (9.1), we would have
or
Differentiating both sides with respect to t, we’d have
Setting t to 0, this would say
This is a well-known differential equation, whose solution is
R(t + u)]/R(t) = R(u) (9.7)
R(t + u) = R(t)R(u) (9.8)
R′(t + u) = R′(t)R(u) (9.9)
R′(u) = R′(0)R(u) (9.10)
R(u) = e −cu
which is exactly 1 minus the cdf for an exponentially distributed random variable.
9.1.2 Example: “Nonmemoryless” Light Bulbs
(9.11)
Suppose the lifetimes in years of light bulbs have the density 2t/15 on (1,4), 0 elsewhere. Say I’ve
been using bulb A for 2.5 years now in a certain lamp, and am continuing to use it. But at this
time I put a new bulb, B, in a second lamp. I am curious as to which bulb is more likely to burn
out within the next 1.2 years. Let’s find the two probabilities.
For bulb A:
For bulb B:
P (L > 3.7|L > 2.5) =
P (X > 1.2) =
4
1.2
P (L > 3.7)
= 0.24 (9.12)
P (L > 2.5)
2t/15 dt = 0.97 (9.13)