Download Handout2 File

Random Variables
Random variables
Sometimes it makes sense to assign a value to an event:
A drunkard making a random walk might be more interested in its
position than in the direction and order of the steps he took.
A gambler might be more interested in his current capital than in
the series of colours already shown at the roulette table.
If you play Settlers of Catan or Monopoly, you are probably more
interested in the sum of the outcomes of the two dice than in the
separate outcomes (unless you suspect that the dice are not fair).
Probability III
Random Variables
Recall that the Borel σ-algebra on R, B is the smallest set
generated by the open (or closed) intervals of R.
Definition
Given a sample space (Ω, F), a random variable is a function
X : Ω → R with the property that the set {ω ∈ Ω : X (ω) ∈ B}
belongs to F for each Borel set B ∈ B, where B is the Borel
σ-algebra on R.
We say that X is F-measurable.
Abuse of notation: {X ∈ B} := {ω ∈ Ω : X (ω) ∈ B} and
{X ≤ x} := {ω ∈ Ω : X (ω) ≤ x}. Furthermore,
P(X ∈ B) := P({X ∈ B}).
Probability III
Random Variables
Proposition
X is F-measurable if and only if {X ≤ x} := {ω ∈ Ω : X (ω) ≤ x}
belongs to F.
Proof:
∞
(−∞, x] = [∪∞
n=1 (x − n, x)] ∪ [∩n=1 (x − 1/n, x + 1/n)],
so, if X is F measurable then {ω ∈ Ω : X (ω) ∈ (−∞, x]}
belongs to F
Suppose that {ω ∈ Ω : X (ω) ∈ (−∞, x]} belongs to F, then
{ω ∈ Ω : X (ω) ∈ (a, b)} =
{ω ∈ Ω : X (ω) ∈ (−∞, a]}c ∩[∪∞
n=1 {ω ∈ Ω : X (ω) ∈ (∞, b−1/n]}]
belongs to F, and therefore for every element B of the
generating set of B, {ω ∈ Ω : X (ω) ∈ B} belongs to F
Probability III
Random Variables
Up to now we did not assign a probability measure to the random
variable X yet. However, there is one natural candidate:
Definition
The distribution measure µX of the random variable X is the
probability measure on (R, B) defined by µX (B) = P(X ∈ B) for
Borel sets B ∈ B.
Note that for disjoint sets B1 , B2 , · · · ∈ B
∞
∞
µX (∪∞
i=1 Bi ) = P(X ∈ ∪i=1 Bi ) = P(∪i=1 {X ∈ Bi })
∞
∞
X
X
=
P(X ∈ Bi ) =
µX (Bi )
i=1
Obviously, µX (R) = P(X ∈ R) = P(Ω) = 1.
So, µX is indeed a probability measure.
Probability III
i=1
Random Variables
Example
A coin is tossed 10 times. Heads appears with probability p
Ω = {ω = (ω1 , · · · , ω10 ); ωi ∈ {H, T }}
F is the power set of Ω
P
X (ω) = 10
1 i = H), i.e., X is the number of heads
i=1 1(ω
P
n
P(ω) = p (1 − p)10−n , where n = 10
1 i = H)
i=1 1(ω
10
X
P(X = n) = |{ω ∈ Ω :
1(ω
1 i = H) = n}|p n (1 − p)10−n
i=1
10 n
=
p (1 − p)10−n
n
So X is Bin(10, n) distributed
Probability III
Random Variables
Distribution function
Definition
The distribution function of the random variable X is the function
FX : R → [0, 1] given by
FX (x) = P(X ≤ x) := µX ((−∞, x])
The distribution function FX , determines µX , since intervals of the
form (−∞, x] generate the Borel σ-algebra B.
Probability III
Random Variables
Some properties of the distribution function F (x) := FX (x)
lim F (x) = 0 and lim F (x) = 1
x→−∞
x→∞
Proof: Let Bn = {X ≤ −n}. the B1 , B2 , · · · is a decreasing
sequence of events with the empty set as limit. We saw in the
previous lecture that this implies that P(Bn ) → P(∅) = 0.
The second statement follows in a similar fashion.
if x < y then F (x) ≤ F (y ), because F (y ) = P(X ≤ y ) =
P(X ≤ x) + P(x < X ≤ y ) ≥ P(X ≤ x) = F (x).
F (x) is right continuous, since {X ≤ x + 1/n} is a decreasing
sequence of events with limit {X ≤ x}.
By monotonicity of F (x) the claim follows.
Probability III
Random Variables
Discrete random variables
X is called discrete if it takes values in some countable (finite or
infinite) subset {x1 , x2 , · · · } of RPsuch that its distribution measure
can be represented as µX (B) = xi ∈B pX (xi ) for B ∈ B and some
function pX : {x1 , x2 , · · · } → [0, 1].
Examples
Bernoulli(p) distribution p(1) = 1 − p(0) = p
Binomial(n, p) distribution p(k) = kn p k (1 − p)n−k for
k = 0, 1, · · · , n
Poisson(λ) distribution p(k) =
λk −λ
k! e
for k ∈ N ∪ {0}
Geometric(p) distribution p(k) = p(1 − p)k−1 for k ∈ N \ {0}
r
k−r
Negative binomial(r , p) distribution p(k) = k−1
r −1 p (1 − p)
for k ∈ N ∩ [r , ∞)
Probability III
Random Variables
Continuous random variables
X is called continuous if Rits distribution measure can be
represented as µX (B) = B fX (x)dx for B ∈ B and some integrable
function fX : R → [0, ∞).
Examples
Uniform(a, b) distribution f (x) = 1/(b − a), for a < x < b
Exponential(λ) distribution f (x) = λe −λx for x ≥ 0
2
Normal(µ,
√ σ ) distribution
f (x) = ( 2πσ 2 )−1 exp(−(x − µ)2 /(2σ 2 )) for x ∈ R
Gamma(λ, t) distribution
(x) = (Γ(t))−1 λt x t−1 e −λx for
R ∞ ft−1
x ≥ 0, where Γ(t) = 0 x e −x dx
Cauchy distribution f (x) =
Probability III
1
π(1+x 2 )
for x ∈ R
Random Variables
side remarks
A random variable might be a mixture of continuous and
discrete random variables
Recreational: There exist also random variables, which are
neither continuous nor discrete (singular random variables).
An example is the random variable with distribution function
the Cantor function.
This distribution has no atoms (The distribution function is
continuous), and it has derivative 0 at any point not in the
Cantor set (which is uncountable). In the Cantor set it has no
derivative.
For more information:
http://en.wikipedia.org/wiki/Cantor_function
Probability III
Random Variables
Proposition
Let the σ-algebra A be generated by a finite partition
P = {A1 , · · · An }. Then the function P
Y is A measurable if and
only if Y may be written as Y (ω) = ni=1 yi1(ω
1 ∈ Ai ) for some
constants y1 , y2 , · · · , yn .
That is, Y is constant on each element of P.
Proof:
Assume that Y is A measurable. For given Ai , choose some
ωi ∈ Ai , and set yi = Y (ωi ).
Ãi = {ω : Y (ω) = yi } ∈ A.Since Y is A-measurable,
it follows that Ãi is the union of some sets in P and it
contains the whole of Ai (the smallest set in A containing ωi ).
The other implication follows since for any Borel set B
{ω ∈ Ω : Y (ω) ∈ B} can be written as the union of sets in P
such that their yi ’s are in B.
Probability III
Random Variables
Example
Ω = [0, 1], B is the Borel σ-algebra on Ω.
P1 = {[0, 1/4], (1/4, 1/2], (1/2, 3/4], (3/4, 1]}
P2 = {[0, 1/2], (1/2, 1]}
Y (x) = min{n ∈ N : n ≥ 4x} for x ∈ [0, 1]
Pi generates Ai for i = 1, 2
Y is A1 measurable, but not A2 -measurable.
Probability III
Random Variables
Definition
The σ-algebra generated by the random variable X is the smallest
σ-algebra A such that X is A measurable.
Example: 3 coin tosses Ω = {(ω1 , ω2 , ω3 ), ωi ∈ {H, T }}
X (ω) is the number of heads.
The smallest σ-algebra containing
A0 = {(T , T , T )}
A1 = {(T , T , H), (T , H, T ), (H, T , T )}
A2 = {(T , H, H), (H, T , H), (H, H, T )}
A3 = {(H, H, H)}
is the σ-algebra generated by X
Probability III
Random Variables
Extra exercises
Solve itens a) and c).
1)(Exam 2008)
Let Ω = {1, . . . , 6} denote the sample space when a dice is rolled
once. Define X(ω) = 1{ω∈{1,2}} and Y(ω)= 1{ω∈{2,3}} .
a) Derive the smallest σ-algebra F that makes X a measurable
rando variable.
b) Introduce the uniform probability measure P and calculate
E (Y | F).
c )Derive the smallest σ-algebra G that makes X and Y
measurable random variables.
Probability III
Random Variables
Extra exercises
1)Show that if X1 , X2 , . . . are random variables (i.e. F-measurable
functions), then infn Xn (ω) and supn Xn are random variables
Hint: By the proposition seen in class (and here) we only need to
check that
{supn Xn ≤ x} ∈ F for x ∈ R.
Probability III