Advanced High-School Mathematics

SECTION 6.2 Continuous Random Variables 359

onk. This says that during the game we don’t “age”; our probability
of dying at the next stage doesn’t increase with age (k).
We now turn this process into a continuous process, where we can
die at any timet≥0 and not just at integer times. We want the process
to enjoy essentially the same condition as the geometric, namely that
if X now denotes the present random variable, then the conditional
probability P(X = t+τ|X ≥ t) should depend on τ but not on t.
In analogy with the above, this conditional probability represents the
probability of living to timet+τ, given that we have already livedt
units of time.
We letf represent the density function ofX; the above requirement
says that

∫t+τ

∫t∞ f(s)ds

t f(s)ds

= function ofτ alone. (∗)

We denote byF an antiderivative off satisfyingF(∞) = 0.^18 There-
fore,

1 =

∫∞

0

f(s)ds = F(s)

∣∣

∣∣

∣

∞

0

=−F(0),

and soF(0) =−1.

Next, we can write (*) in the form

F(t+τ)−F(t)

−F(t)

= g(τ),

for some functiong. This implies that the quotient

F(t+τ)

F(t)

doesn’t

depend ont. Therefore, the derivative with respect totof this quotient
is 0:

F′(t+τ)F(t)−F(t+τ)F′(t)

F(t)^2

= 0,

forcing

(^18) Since
∫∞
0
f(s)ds= 1, we see thatFcannot be unbounded at∞.