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.

Share and discuss your solutions below.

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

Solution 2

Write it here.

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