1. (15% of Total)

This question is a set of short-answer questions (no proofs):

(a) (5%)

State the definition of a Probability Space.

You can see the definition of a Probability Space.

(b) (5%)

State the definition of a random variable; use notation from your answer in part (a).

Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

August 2003

A random variable $\mathbf{X}$ is a process of assigning a number $\mathbf{X}\left(\xi\right)$ to every outcome $\xi$ . The result function must satisfy the following two conditions but is otherwise arbitrary:

1. The set $\left\{ \mathbf{X}\leq x\right\}$ is an event for every $x$ .

2. The probabilities of the events $\left\{ \mathbf{X}=\infty\right\}$ and $\left\{ \mathbf{X}=-\infty\right\}$ equal 0.

Given $\left(\mathcal{S},\mathcal{F},\mathcal{P}\right)$ , a random variable $\mathbf{X}$ is a mapping from $\mathcal{S}$ to the real line. $\mathbf{X}:\mathcal{S}\rightarrow\mathbb{R}$ .

(c) (5%)

State the Strong Law of Large Numbers.