
28 The empirical distribution function of duration of life
for any positive value of the parameter λ . We will select a specific
value later. Let us use the Markov inequality
P{e
λ(ξ
n
−np)
≥ e
nλε
} ≤
Ee
λ(ξ
n
−np)
e
nλε
.
For binomial ξ it is, however, easy to calculate the expected value
Ee
λ(ξ
n
−np)
on the right (for this it is sufficient to use the binomial for-
mula):
Ee
λξ
n
= (1 + p(e
λ
−1))
n
.
Using the inequality (1 + x)
n
< e
nx
for this last expression we finally
obtain
P
ξ
n
n
−p ≥ε
≤ e
np(e
λ
−1)
e
−nλ p−nλ ε
= e
np(e
λ
−1−λ)−nλ ε
.
Not for all values of λ does the right-hand side produce a useful in-
equality. Hence, it is important to choose λ that minimizes the expo-
nent. Dif