Page 305 - Elements of Distribution Theory

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052184472Xc09 CUNY148/Severini June 2, 2005 12:8

9.7 Approximation of Sums 291

with error

r
P(x) f (x) dx. (9.9)

m
Example 9.15 (Discrete uniform distribution). Let X denote a discrete random variable
with a uniform distribution on the set {0, 1,..., m} for some positive integer m; hence,

1
Pr(X = j) = , j = 0,..., m.
m + 1
Let f denote a bounded, real-valued function on [0, 1] and consider the expected value
E[ f (X/m)]. Let U denote an absolutely continuous random variable with a uniform distri-
bution on (0, 1). Here we consider the approximation of E[ f (X/m)] by E[ f (U)] for large

m.We assume that f is differentiable and that f satisfies the Lipschitz condition
| f (s) − f (t)|≤ K|s − t|, s, t ∈ [0, 1]

for some constant K.
Using Theorem 9.17,
m
1
E[ f (X/m)] = f ( j/m)
m + 1
j=0
1 m 1
= f (x/m) dx + [ f (0) + f (1)]
m + 1 0 2(m + 1)
1 m

+ (x − x − 1/2) f (x/m) dx.
m(m + 1) 0
Changing the variable of integration,

1 m m 1 m
f (x/m) dx = f (u)du = E[ f (U)].
m + 1 0 m + 1 0 m + 1

Note that, for j ≤ x < j + 1,
x − x − 1/2 = x − j − 1/2;

hence,

m m−1 j+1

(x − x − 1/2) f (x/m) dx = (x − j − 1/2) f (x/m) dx
0 j=0 j
1
m−1
2 u + j + 1/2
= uf
du.
1 m
j=0 − 2
Since
1

2
cu du = 0
1
−
2