Download Lecture 14: Cumulative Distribution Functions and Continuous

Lecture 14: Cumulative Distribution Functions and Continuous Random Variables
1. Properties of the CDF
Proposition: Let X be a real-valued random variable (not necessarily discrete) with cumulative distribution function (CDF) F (x) = P(X ≤ x). Then
a) F is non-decreasing, i.e., if x < y, then F (x) ≤ F (y).
b) limx→∞ F (x) = 1.
c) limx→−∞ F (x) = 0.
d) F is right-continuous, i.e., for any x and any decreasing sequence (xn , n ≥ 1), that converges
to x, limn→∞ F (xn ) = F (x).
Proof:
a) If x < y, then {X ≤ x} ⊂ {X ≤ y}, which implies that F (x) ≤ F (y).
b) If (xn , n ≥ 1) is an increasing sequence such that xn → ∞, then the events En = {X ≤ xn }
form an increasing sequence with
{X < ∞} =
∞
[
En .
n=1
It follows from the continuity properties of probability measures (Lecture 2) that
lim F (xn ) = lim P(En ) = P(X < ∞) = 1.
n→∞
n→∞
c) Likewise, if (xn , n ≥ 1) is a decreasing sequence such that xn → −∞, then the events
En = {X ≤ xn } form a decreasing sequence with
∅=
∞
\
En .
n=1
In this case, the continuity properties of measures imply that
lim F (xn ) = lim P(En ) = P(∅) = 0.
n→∞
n→∞
d) If xn , n ≥ 1 is a decreasing sequence converging to x, then the sets En = {X ≤ xn } also form
a decreasing sequence with
∞
\
{X ≤ x} =
En .
n=1
Consequently,
lim F (xn ) = lim P(En ) = P{X ≤ x} = F (x).
n→∞
n→∞
Some other notable properties of the CDF are:
1
i) P(a < X ≤ b) = F (b) − F (a) for all a < b.
ii) Another application of the continuity property shows that the probability that X is strictly
less than x is equal to
P(X < x) =
lim P(X ≤ x − 1/n) =
n→∞
lim F (x − 1/n).
n→∞
In other words, P(X < x) is equal to the left limit of F at x, which is often denoted F (x−).
However, notice that in general this limit need not be equal to F (x):
F (x) = P(X < x) + P(X = x),
and so P(X < x) = F (x) if and only if P(X = x) = 0.
iii) The converse to this proposition is also true: namely, if F : R → [0, 1] is a function satisfying
properties (a) - (d), then F is the CDF of some random variable X.
2.) Continuous Random Variables
Definition: A real-valued random variable X is said to be a continuous random variable if
there is a non-negative function f : R → [0, ∞) such that
Z b
P(a ≤ X ≤ b) =
f (x)dx
a
for all a < b. The function f is called the probability density function of X.
Remarks:
Notice that
Z
∞
1 = P(−∞ < X < ∞) =
f (x)dx,
−∞
and so the integral of a density function over the entire real line must be equal to 1.
Since
Z
P(X = x) =
x
f (y)dy = 0,
x
the probability that a continuous RV will assume any particular value is 0.
There is a close connection between the density of a random variable and its CDF. First, observe
that
Z x
F (x) = P(X ≤ x) =
f (y)dy.
∞
Since the right-hand side is a differentiable function of x, it follows that F (x) is also differentiable
and that
F 0 (x) = f (x),
2
i.e., the density is the derivative of the cumulative distribution function.
Example: If X is a continuous RV with CDF Fx and density fx , find the CDF and the density
function of Y = 2X.
Solution: The CDF of Y is equal to
FY (x) = P(Y ≤ x) = P(2X ≤ x) = P(X ≤ x/2) = FX (x/2),
and differentiation with respect to x gives
fY (x) =
1
d
FY (x) = fX (x/2).
dx
2
3