Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

January 2001

# Part 2

(a) (7 pts)

Let $A$ and $B$ be statistically independent events in the same probability space. Are $A$ and $B^{C}$ independent? (You must prove your result).

(b) (7 pts)

Can two events be statistically independent and mutually exclusive? (You must derive the conditions on A and B for this to be true or not.)

(c) (6 pts)'

State the Axioms of Probability.

# Solution 1 (retrived from here)

(a)

$P\left(A\right)=P\left(A\cap\left(B\cup B^{C}\right)\right)=P\left(\left(A\cap B\right)\cup\left(A\cap B^{C}\right)\right)=P\left(A\cap B\right)+P\left(A\cap B^{C}\right)=P\left(A\right)P\left(B\right)+P\left(A\cap B^{C}\right).$

$P\left(A\cap B^{C}\right)=P\left(A\right)-P\left(A\right)P\left(B\right)=P\left(A\right)\left(1-P\left(B\right)\right)=P\left(A\right)P\left(B^{C}\right).$

$\therefore A\text{ and }B^{C}\text{ are independent. }$

(b)

If $P\left(A\right)=0$ or $P\left(B\right)=0$ , then A and B are statistically independent and mutually exclusive. Prove this:

Without loss of generality, suppose that $P\left(A\right)=0$ . $0=P\left(A\right)\geq P\left(A\cap B\right)\geq0\Longrightarrow P\left(A\cap B\right)=0\qquad\therefore\text{mutually excclusive}.$

$P\left(A\cap B\right)=0=P\left(A\right)P\left(B\right)\qquad\therefore\text{statistically independent.}$

(c)

Axioms of probability=

• The probability measure $P\left(\cdot\right)$ corresponding to $S$ and $F\left(S\right)$ is the assignment of a real number $P\left(A\right)$ to each $A\in F\left(S\right)$ satisfying following properties. Axioms of probability:

1. $P\left(A\right)\geq0$ , $\forall A\in F\left(S\right)$ .

2. $P\left(S\right)=1$ .

3. If $A_{1}$ and $A_{2}$ are disjoint events, then $P\left(A_{1}\cup A_{2}\right)=P\left(A_{1}\right)+P\left(A_{2}\right)$ . If $A_{1},A_{2}\in F\left(S\right)$ and $A_{1}\cap A_{2}=\varnothing$ , then $A_{1}$ and $A_{2}$ are disjoint events.

4. If $A_{1},A_{2},\cdots,A_{n},\cdots\in F\left(S\right)$ is a countable collection of disjointed events, then $P\left(\bigcup_{i=1}^{\infty}A_{i}\right)=\sum_{i=1}^{\infty}P\left(A_{i}\right)$ .

$P\left(\cdot\right)$ is a set function. $P\left(\cdot\right):F\left(S\right)\rightarrow\mathbf{R}$ .

• If you want to talk about the probability of a single output $\omega_{0}\in S$ , you do so by considering the single event

Write it here.

## Alumni Liaison

Recent Math PhD now doing a post-doctorate at UC Riverside.

Kuei-Nuan Lin