MATH 180A Lecture 7

Theorem: Let $A,B$ be two events in a probability space. Then

are independent are independent.

Proof. Assume independent.

Remember




Shows that are independent

Example. Suppose are independent events in a probability space w/ and . What is the probability that exactly one of occurs?

Mutual independence

A sequence of events in a probability space are mutually independent if any collection of events

For example, are mutually independent if

Example. I toss a fair coin 3 times. Consider the events
= {exactly 1 tails in 1st and 2nd toss} = {THH,THT,HTH,HTT},
= {exactly 1 tails in 2nd or 3rd toss} = {HTH,TTH,HHT,THT},
= {exactly 1 tails in 1st or 3rd toss} = {THH,TTH,HHT,HTT},
Are mutually independent?
,
,
,
they are pairwise independent

not mutually independent

Section 1.5

Random Variables

Random Variable

A real-valued random variable is a function where is the sample space of some probability space, and is measurable.

’s are intervals

Examples.

Fair coin toss: $\Omega =\{H,T\}$. Define $X:\Omega \to \mathbb{R}$

Roll a fair die twice: $\Omega =\{1,2,3,4,5,6{\}}^2$. Define $S:\Omega \to \mathbb{R}$

is the sum of the 2 rolls

Chooes a point uniformly on unit disk $\Omega =\{(x,y)\in {\mathbb{R}}^2:x^2+y^2+2\}$. Define $x:\Omega \to R$

distance to center of when dart hits.

Distributions

Probability Distribution

Let ( be a probability space a random variable. The probability distribution of is the probability measure on
for all .

Examples.

Fair coin toss $\Omega =\{H,T\}$. Define $X:\Omega \to \mathbb{R}$ by $X(H)=\pi ,X(T)=-\sqrt{2}$. What is the distribution of $X$?

for every