Measure, Integration & Real Analysis, 2021a

396 Chapter 12 Probability Measures
Weak Law of Large Numbers
Families of random variables all of which look the same in terms of their distribution
functions get a special name, as we see in the next definition.
12.35 Definition identically distributed; i.i.d.
Suppose (Ω, F, P) is a probability space.
• A family of random variables on (Ω, F) is called identically distributed if
all the random variables in the family have the same distribution function.
• More specifically, a family {X k } k∈Γ of random variables on (Ω, F) is called
identically distributed if
for all j, k ∈ Γ.
P(X j ≤ s) =P(X k ≤ s)
• A family of random variables that is independent and identically distributed
is said to be independent identically distributed, often abbreviated as i.i.d.
12.36 Example family of random variables for decimal digits is i.i.d.
Consider the probability space ([0, 1], B, P), where B is the collection of Borel
subsets of the interval [0, 1] and P is Lebesgue measure on ([0, 1], B). Fork ∈ Z + ,
define a random variable X k : [0, 1] → R by
X k (ω) =k th -digit in decimal expansion of ω,
where for those numbers ω that have two different decimal expansions we use the
one that does not end in an infinite string of 9s.
Notice that P(X k ≤ π) =0.4 for every k ∈ Z + . More generally, the family
{X k } k∈Z + is identically distributed, as you should verify.
The family {X k } k∈Z + is also independent, as you should verify. Thus {X k } k∈Z +
is an i.i.d. family of random variables.
Identically distributed random variables have the same expectation and the same
standard deviation, as the next result shows.
12.37 identically distributed random variables have same mean and variance
Suppose (Ω, F, P) is a probability space and {X k } k∈Γ is an identically distributed
family of random variables in L 2 (P). Then
for all j, k ∈ Γ.
EX j = EX k and σ(X j )=σ(X k )
Measure, Integration & Real Analysis, by Sheldon Axler