Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

January 2004

1. (30 pts.)

This question consists of two separate short questions relating to the structure of probability space:

(a)

Assume that $\mathcal{S}$ is the sample space of a random experiment and that $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ are $\sigma$ -fields (valid event spaces) on $\mathcal{S}$ . Prove that $\mathcal{F}_{1}\cap\mathcal{F}_{2}$ is also a $\sigma$ -field on $S$ .

(b)

Consider a sample space $\mathcal{S}$ and corresponding event space $\mathcal{F}$ . Suppose that $P_{1}$ and $P_{2}$ are both balid probability measures defined on $\mathcal{F}$ . Prove that $P$ defined by $P\left(A\right)=\alpha_{1}P_{1}\left(A\right)+\alpha_{2}P_{2}\left(A\right),\qquad\forall A\in\mathcal{F}$ is also a valid probability measure on $\mathcal{F}$ if $\alpha_{1},\;\alpha_{2}\geq0$ and $\alpha_{1}+\alpha_{2}=1$ .

• Because $P_{1}$ and $P_{2}$ are valid probability measures, we know that they satisfy the axioms of probability:

1. $P_{1}\left(A\right)\geq0$ and $P_{2}\left(A\right)\geq0$ , $\forall A\in\mathcal{F}\left(\mathcal{S}\right)$ .

2. $P_{1}\left(\mathcal{S}\right)=1$ and $P_{2}\left(\mathcal{S}\right)=1$ .

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

4. If $A_{1},A_{2},\cdots,A_{n},\cdots\in\mathcal{F}\left(\mathcal{S}\right)$ is countable collection of disjoint events, then $P_{1}\left(\cup_{i=1}^{\infty}A_{i}\right)=\sum_{i=1}^{\infty}P_{1}\left(A_{i}\right)$ and $P_{2}\left(\cup_{i=1}^{\infty}A_{i}\right)=\sum_{i=1}^{\infty}P_{2}\left(A_{i}\right)$ .

• Now, we check each condition to become a valid probability measure:

1. $P\left(A\right)=\alpha_{1}P_{1}\left(A\right)+\alpha_{2}P_{2}\left(A\right)\geq0 , \forall A\in\mathcal{F}\left(\mathcal{S}\right)$ .

$\because\alpha_{1}\geq0,\;\alpha_{2}\geq0,\; P_{1}\left(A\right)\geq0,\text{ and }P_{2}\left(A\right)\geq0$ .

2. $P\left(S\right)=\alpha_{1}P_{1}\left(A\right)+\alpha_{2}P_{2}\left(A\right)=\alpha_{1}+\alpha_{2}=1$ .

3. If $A_{1}$ and $A_{2}\in\mathcal{F}\left(\mathcal{S}\right)$ are disjoint events, then $P\left(A_{1}\cup A_{2}\right)=\alpha_{1}P_{1}\left(A_{1}\cup A_{2}\right)+\alpha_{2}P_{2}\left(A_{1}\cup A_{2}\right)=\alpha_{1}\left\{ P_{1}\left(A_{1}\right)+P_{1}\left(A_{2}\right)\right\} +\alpha_{2}\left\{ P_{2}\left(A_{1}\right)+P_{2}\left(A_{2}\right)\right\} $$=\alpha_{1}P_{1}\left(A_{1}\right)+\alpha_{2}P_{2}\left(A_{1}\right)+\alpha_{1}P_{1}\left(A_{2}\right)+\alpha_{2}P_{2}\left(A_{2}\right)=P\left(A_{1}\right)+P\left(A_{2}\right). 4. If A_{1},A_{2},\cdots,A_{n},\cdots\in\mathcal{F}\left(\mathcal{S}\right) is countable collection of disjoint events, then P\left(\cup_{i=0}^{\infty}A_{i}\right)=\alpha_{1}P_{1}\left(\cup_{i=0}^{\infty}A_{i}\right)+\alpha_{2}P_{2}\left(\cup_{i=0}^{\infty}A_{i}\right)=\alpha_{1}\sum_{i=1}^{\infty}P_{1}\left(A_{i}\right)+\alpha_{2}\sum_{i=1}^{\infty}P_{2}\left(A_{i}\right)$$ =\sum_{i=1}^{\infty}\left\{ \alpha_{1}P_{1}\left(A_{i}\right)+\alpha_{2}P_{2}\left(A_{i}\right)\right\} =\sum_{i=1}^{\infty}P\left(A_{i}\right).$

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