Random Variables and Stochastic Processes 0903720 Dr Ghazi
Random Variables and Stochastic Processes – 0903720 Dr. Ghazi Al Sukkar Email: ghazi. alsukkar@ju. edu. jo Office Hours: will be posted soon Course Website: http: //www 2. ju. edu. jo/sites/academic/ghazi. alsukkar Most of the material in these slide are based on slides prepared by Dr. Huseyin Bilgekul http: //faraday. ee. emu. edu. tr/ee 571/ EE 720 1
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 1 1. 7 Random Selection of Points from Intervals EE 720 2
Set Review I • EE 720 3
Set Review II • For more: http: //www. rapidtables. com/math/symbols/Set_Symbols. htm EE 720 4
1. 1: Introduction Overview: An event may or may not occur • What’s Random? Occurrence of an event is inevitable • What’s Certain? An event can never occur • 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 720 5
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 NA is the number of outcomes that belongs to A ⊂ S. P(A) =NA/N EE 720 7
1. 2: Sample Space and Events • Sample Space: The set of all possible outcomes, denoted by S note: A sample space can be discrete or continuous. q Sample points: q Events: Certain subsets of S are referred to as events. Sample Space b a d k f e c h g j m l i n EE 720 o p Sample point Event 8
Examples q Example 1. 1: For the experiment of tossing a coin once, what is the sample space S ? q 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? EE 720 9
Examples q 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 505. 5 hours. EE 720 10
Examples q 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={ } } } EE 720 11
Examples q 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: EE 720 12
Examples q 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={ } EE 720 13
Occurrence of an Event q 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, sd} have occurred. Spade Heart Diamond Club EE 720 14
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