Advanced High-School Mathematics

360 CHAPTER 6 Inferential Statistics

F′(t+τ)F(t) =F(t+τ)F′(t).

But this can be written as

d

dt

lnF(t+τ) =

d

dt

lnF(t),

forcing

F(t+τ) =−F(t)F(τ),

for alltandτ. Finally, if we differentiate both sides of the above with
respect totand then sett= 0, we arrive at

F′(τ) =−F′(0)F(τ),

which, after setting λ = F′(0), easily implies that F(t) = −e−λt for
allt≥0. Sincef is the derivative ofF, we conclude finally, that the
density function of the exponential distribution must have the form

f(t) = λe−λt, t≥ 0.

Very easy integrations show that

E(X) =

1

λ

and Var(X) =

1

λ^2

.

The exponential distribution is often used in reliability engineering
to describe units having a constant failure rate (i.e., age independent).
Other applications include

• modeling the time to failure of an item (like a light bulb; see
  Exercise 2, below). The parameter λ is often called the failure
  rate;


• modeling the time to the next telephone call;


• distance between roadkill on a given highway;