Random Variables and Stochastic Processes – Dr. Ghazi Al Sukkar Office Hours: will be posted soon Course Website: EE 7201 Most of the material in these slide are based on slides prepared by Dr. Huseyin Bilgekul /

EE 7202 PART 1 Axioms of Probability 1. Set review 1.1 Introduction 1.2 Sample Space and Events 1.3 Axioms of Probability 1.4 Basic Theorems 1.5 Continuity of Probability Function 1.6 Probabilities 0 and Random Selection of Points from Intervals

Set Review I EE 7203

Set Review II EE 7204 For more:

1.1: Introduction Overview: What’s Random? What’s Certain? What’s Impossible? Examples ： 1.Disintegration of a given atom of radium 2.Finding no defect during inspection of a microwave oven 3.Orbit satellite in space is at a certain position 4.An object travels faster than light 5.A thunderstorm flashes of lighting precede any thunder echoes EE 7205 An event may or may not occur Occurrence of an event is inevitable An event can never occur

Random experiment Random experiment: its outcome, for some reason, cannot be predicted with certainty. Examples: throwing a die, flipping a coin and drawing a card from a deck.

Relative Frequency Interpretation Definition ： (Probability based on experiment): The number of times the experiment is performed is n, and n A is the number of times, where the outcome belongs to A ⊂ S. The probability P of event A is defined as Dilemmas ： 1.Can not be computed since n , only approximation 2.Does the limit of n A /n exist? 3.Probabilities that are based on our belief and knowledge are not justifiable.  The probability that the price of oil will be raised in the next six months is 60%.  The probability that it will snow next Christmas is 30% Classical Definition: N is the total number of outcomes, and N A is the number of outcomes that belongs to A ⊂ S. P(A) =N A /N EE 7207

1.2: Sample Space and Events Sample Space ： EE 7208 The set of all possible outcomes, denoted by S note: A sample space can be discrete or continuous.  Sample points ：  Events ： Certain subsets of S are referred to as events. a b e f g k i pd m c h j l n o Sample Space Sample point Event

Examples EE 7209  Example 1.1 ： For the experiment of tossing a coin once, what is the sample space S ?  Example 1.2 ： Suppose that an experiment consists of two steps. First a coin is flipped. If the outcome is tails, a die is tossed. If the outcome is head, the coin is flipped again. What is the sample space S ? What is the event of heads in the first flip of the coin? What is the event of an odd outcome when the die is tossed?

Examples EE  Example 1.3 ： Consider measuring the lifetime of a light bulb. Since any nonnegative real number can be considered as the lifetime of the light bulb (in hours), the sample space S is S = {x ： x  0 }. The event E = {x ： x  100 } is the event that the light bulb lasts at least 100 hours. The event F = {x ： x  1000 } is the event that it lasts at most 1000 hours. The event G = {505.5} is the event that it lasts exactly hours.

Examples EE  Example 1.4 ： Suppose that a study is being done on all families with one, two, or three children. Let the outcome of the study be the genders of the children in descending order of their ages. What is the sample space S ? What is the event F where the eldest child is a boy? What is the event G where families with exactly 2 girls? Ans ： S = { } F = { } G = { }

Examples EE  Example 1.5 ： A bus with a capacity of 34 passengers stops at a station some time between 11:00 AM and 11:40 AM every day. What is the sample space of the experiment, consists of counting the number of passengers on the bus and measuring the arrival time of the bus? Ans: What is the event? Ans:

Examples EE  Example 1.6 ： (Round-off Error) Suppose that each time Jay charges an item to his credit card, he will round the amount to the nearest dollar in his records. Therefore, the round-off error, which is the true value charged minus the amount recorded, is random, with the sample space S = { } The event of rounding off at most 3 cents in a random charge is given by S = { }

Occurrence of an Event EE  If the outcome of an experiment belongs to an event E, we say that the event E has occurred. If we draw two cards from an ordinary deck of 52 cards and observe that one is a spade and the other a heart, all of the following events have occurred. {sh}, {sh, dd}, {cc, dh, sh}, {hc, sh, ss, hh}, {cc, hh, sh, dd} But none of the events {dh, sc}, {dd}, {cc, hh, ss}, {hd, hc, dc, sc, sd} have occurred. Spade Diamond Heart Club