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Sections 1.5, 1.6, and 1.7 lecture notes Math331, Fall 2008 Instructor: David Anderson
Section 1.5: Continuity of Probability Functions
P (·) is a function from P(S) to R. So what does continuity mean?
Recall: A function f : R → R is continuous at a point c if for all x
lim x→c f (x) = f (c).
Equivalently, f is continuous on R if and only if for every convergent sequence {xn},
nlim→∞ f^ (xn) =^ f^ ( lim n→∞ xn).
We will use this as method to define continuity.
A sequence of subsets {En} is called increasing if
E 1 ⊂ E 2 ⊂ · · · En ⊂ En+1 ⊂ · · ·
and is called decreasing if
E 1 ⊃ E 2 ⊃ · · · En ⊃ En+1 ⊃ · · ·
Note that for an increasing sequence
En =
⋃^ n
i=
En =⇒ (^) nlim→∞ En =
⋃^ ∞
i=
En.
and that for a decreasing sequence
En =
⋂^ n
i=
En =⇒ lim n→∞
En =
⋂^ ∞
i=
En.
Theorem 1 (Continuity of Prob. Funcs). For any increasing or decreasing sequence of events, {En}, lim n→∞
P (En) = P ( lim n→∞
En).
Proof. Let Ei be increasing. Let Fi = Ei −Ei− 1. Then the F ’s are mutually exclusive. Why? Thus,
P ( lim n→∞ En) = P (
⋃^ ∞
i=
Ei) = P (
⋃^ ∞
i=
Fi) =
∑^ ∞
i=
P (Fi)
= lim n→∞
∑^ n
i=
P (Fi) = lim n→∞ P (
⋃^ n
i=
Fi) = lim n→∞ P (
⋃^ n
i=
Ei)
= lim n→∞
P (En).
Section 1.6: Probabilities 0 and 1 (and an example for Sec 1.5)
Whole (important) point of section: If E and F are events with probabilities 1 and 0, respectively, it is not correct to say that E is the sample space, S, and F is the empty set ∅.
Example 1 (Selecting a point at random from the interval (0, 1)). How do you do it? Every point has a decimal representation! So, for example, .4927848548381.... Each goes on forever (but may repeat). So we can pick a number by using the sample space
S = {(i 1 , i 2 , i 3 ,... ) : ik ∈ { 0 ,... , 9 }}.
Consider the probability of selecting 1/3 =. 3333333 .... Let An be the event that we have chosen a 3 for the first n digits. Then,
A 1 ⊃ A 2 ⊃ A 3 ⊃ · · ·.
Also, P (An) = P (An− 1 )/10 since there is a 1 in 10 chance of picking a 3 at each step. Because P (A 1 ) = 1/10, we have P (An) = (1/10)n. Also,
i=1 An^ =^ {^1 /^3 }.^ Thus, by continuity theorem
P
= P
i=
An
= lim n→∞ P (An) = lim n→∞
)n = 0.
Also note that
1 = P (S) = P ((S − { 1 / 3 }) ∪ { 1 / 3 }) = P (S − { 1 / 3 }) + P ({ 1 / 3 }) = P (S − { 1 / 3 }).
Trivial to do for any number. Much more surprisingly, if {ai} is a countably infinite collection of points:
P
i=
{ai}
∑^ ∞
i=
P ({ai}) =
∑^ ∞
i=
Read Section 1.7 (not Example 1.21 unless you want to)
Homework: Pg. 34 #’s 1, 3, 4, 5, 10.
