1. (15% of Total)

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

(a) (5%)

State the definition of a Probability Space.

Answer

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).

ECE Ph.D. Qualifying Exam

Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

August 2003



Answer (74p on Papoulis)

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.

Answer (Intuitive definition)

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} $ .

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(c) (5%)

State the Strong Law of Large Numbers.

Answer

You can see the definition of the Strong Law of Large Numbers.

Alumni Liaison

Correspondence Chess Grandmaster and Purdue Alumni

Prof. Dan Fleetwood