Chapter_01 Prob and Random Variables.pptx

Ch_01 Prob and Random Variables.pptx

1. Probability and Random process Chapter 1: Probability and Random Variables section one Abdirahman Farah Ali C .raxmaanfc@ gmail.com 1

2. Class rules 2 Attendance is mandatory . You are responsible for whatever is taught in the lecture . If you miss a class, it is your responsibility to find out about assignment, quizzes and exam. Punctuality is compulsory . You are encouraged to collaborate ( not copy) on assignment problems with your "study buddies.” Respect and listen during period Take notes in lectures Ask questions

3. Electrical engineering students need to study probability and random processes for several reasons: 1. To analyze and design communication systems: Communication systems such as wireless networks, satellite communications, and radar systems use random signals, noise, and interference. Probability and random processes provide the tools to analyze the behavior of these systems and design them to meet specific performance criteria. 2. To analyze and design control systems: Control systems such as feedback control systems, robotics, and industrial automation systems also use random signals and noise. Probability and random processes provide the tools to analyze the stability, performance, and robustness of these systems. 3 . To analyze and design power systems: Power systems such as power grids and renewable energy systems are subject to random fluctuations in supply and demand. Probability and random processes provide the tools to model and analyze these fluctuations and design the systems to maintain stability and reliability. 4. To analyze and design electronic circuits: Electronic circuits such as amplifiers and filters are subject to random noise and fluctuations in their operating parameters. Probability and random processes provide the tools to analyze the impact of these random effects on circuit performance and design the circuits to minimize their impact. 3

4. How important is probability for an electrical engineer? An electrical engineer is likely to deal with "probability" in each of several areas as follows: In the analysis of communications systems, one is frequently concerned with lower signal-to-noise ratios resulting in " bit-errors" , false alarms, etc. There is then a need for careful positioning of thresholds, use of error-correcting codes, et-cetera to insure that the probability of error is kept within acceptable bounds. In the design of circuits and systems , it is often important that the characteristic of some circuit element like its resistance or inductance has to be kept within a certain range in order for the circuit to operate properly, so the probability distribution of this kind of parameter becomes important. In product development and manufacturing planning, uncertainty as to product demand and component availability are frequently of concern, and again, intelligent judgments as to probability are required In summary, probability and random processes are essential tools for electrical engineering students to understand the behavior of complex systems and design them to meet specific performance criteria. 4

5. Probability Probability theory is the branch of mathematics that is concerned with the study of random phenomena. A random phenomenon is one that, under repeated observation, yields different outcomes that are not deterministically predictable. It refers to an event or outcome that cannot be predicted with certainty. Randomness is a fundamental concept in probability theory and statistics, and it is characterized by the fact that the outcome of a random phenomenon cannot be determined ahead of time Probability  refers to how likely an event is to occur. Probability is used in all types of areas in real life including weather forecasting, sports betting, investing, and more. 5

6. Continue.. Probability  means possibility . It is a branch of mathematics that deals with the occurrence of a random event . The value is expressed from zero to one. Probability has been introduced in Mathematics to predict how likely events are to happen.  The meaning of probability is basically the extent to which something is likely to happen. This is the basic probability theory, which is also used in the probability distribution, where you will learn the possibility of outcomes for a random experiment. To find the probability of a single event to occur, first, we should know the total number of possible outcomes. 6

7. Applications Applications Probability theory is applied in everyday life in risk assessment and in trade on financial markets. Governments apply probabilistic methods in environmental regulation , where it is called pathway analysis. Another significant application of probability theory in everyday life is reliability. Many consumer products, such as automobiles and consumer electronics , use reliability theory in product design to reduce the probability of failure. Failure probability may influence a manufacturer's decisions on a product's warranty. The range of applications extends beyond games into business decisions, insurance, law, medical tests, and the social sciences The telephone network, call centers , and airline companies with their randomly fluctuating loads could not have been economically designed without probability theory . 7

8. Continue………. Weather Forecasting: rain , snow, clouds, etc. on a given day in a certain area. Politics: the chances that certain candidates will win various elections. Sales Forecasting: Many : retail companies use probability to predict the chances that they’ll sell a certain amount of goods in a given day, week, or month. Natural Disasters  Traffic Investing 8

9. Probability (Cont.…) Examples of these random phenomena include the number of electronic mail (e-mail) messages received by all employees of a company in one day, the number of phone calls arriving at the university’s switchboard over a given period, the number of components of a system that fail within a given interval , and the number of A’s that a student can receive in one academic year. 9

10. The Terminology of Probability 1. Random Experiments In the study of probability, any process of observation is referred to as an experiment. The results of an observation are called the outcomes of the experiment. An experiment is called a random experiment if its outcome cannot be predicted. Typical examples of a random experiment are the roll of a die, the toss of a coin , or selecting a message signal for transmission from several messages. 10

11. The Terminology of Probability 2. Sample Space: The set of all possible outcomes of a random experiment is called the sample space (or universal set), and it is denoted by S. An element in S is called a sample point. Each outcome of a random experiment corresponds to a sample point. 11

12. 3. A random variable A random variable is a numerical quantity that is generated by a random experiment or process. In probability theory, a random variable is defined as a function that maps the outcomes of a random event to numerical values. For example, consider a coin toss where we assign a value of 1 for heads and 0 for tails. The random variable X can be defined as the outcome of the coin toss, with the possible values of X being 0 or 1. Random variables can be either discrete or continuous. A discrete random variable takes on a finite or countable number of distinct values, while a continuous random variable takes on values in a continuous range. 12

13. Example 1.1 Find the sample space for the experiment of tossing a coin (a) once (b) twice. SOLUTION (a) There are two possible outcomes, heads or tails. Thus S = {H, T) where H and T represent head and tail, respectively. (b) There are four possible outcomes. They are pairs of heads and tails. Thus S = (HH, HT, TH, TT) 13

14. Example 1.2 If you toss 3 coins, “n” is taken as 3. solution Therefore, the possible number of outcomes will be 2 3  = 8 outcomes Sample space for tossing three coins is written as Sample space S = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} 14

15. Example 1.3 A Die is Thrown When a single die is thrown, it has 6 outcomes since it has 6 faces. Therefore, the sample is given as S = { 1, 2, 3, 4, 5, 6} What is probability of getting even number? SOLUTION P(Even no.)=3/6,=1/2,=0.5, 50% 15

16. Probability as the ratio of favorable to total outcomes (classical probability) The probability of an event is computed a PRIORI by counting the number of ways that event can occur and forming the ratio.   The probability of an event occurring is the number in the event divided by the number in the sample space . Again, this is only true when the events are equally likely. P(E) = n(E) / n(S) 16

17. Probability rules All probabilities are between 0 and 1 inclusive PE = 0 ≤ P ≤ 1 The sum of all the probabilities in the sample space is 1 For example, in a coin flip, the probability of heads is 0.5 and the probability of tails is also 0.5 , and the sum of these probabilities is 1 . There are some other rules which are also important. The probability of an event which cannot occur is 0. An event with probability cannot occur, and an event with probability 1 is certain to occur The probability of any event which is not in the sample space is zero. The sum of the probabilities of all possible outcomes in a given event is equal to 1. 17

18. probability rules The probability of an event which must occur is 1. The probability of an event not occurring is one minus the probability of it occurring. P(E') = 1 - P(E) 18

19. Conditional probability Definition : Conditional probability is the probability of an event occurring given that another event has occurred. Mathematically, conditional probability can be represented as P(A|B), which means the probability of event A given that event B has occurred. For example, the probability of it raining today given that the weather forecast predicts rain for today. The conditional probability of an event A given event B, denoted by P(A | B), is defined as Conditional probability P(A|B)= P(A and B) is the probability of both events A and B occurring together, and P(B) is the probability of event B occurring.   19

20. Conditional probability (cont..) the conditional probability of an event B given event A. We have P(A ∩ B)= P(A|B).P(B) P(B ∩A )= P(B|A).P(A) , This equation is often quite useful in computing the joint probability of events. 20

21. EXAMPLE 1.4 A lot of 100 semiconductor chips contains 20 that are defective. Two chips are selected at random, without replacement, from the lot. (a) What is the probability that the first one selected is defective? (b) What is the probability that the second one selected is defective given that the first one was defective? (c) What is the probability that both are defective? 21

22. SOLUTION ( A) Let A denote the event that the first one selected is defective then, by P(A)= =0.2 (B) Let B denote the event that the second one selected is defective. After the first one selected is defective, there are 99 chips left in the lot with 19 chips that are defective. Thus, the probability that the second one selected is defective given that the first one was defective is P(B|A)=   22 (C ) the probability that both are defective is P(A∩B)= P(B|A)P(A) P(A∩B)= ( = 0.0384 ∩ this symbol is know a intersection The probability of the intersection of two events equals the probability that both events occur  

23. Example 1.5 A bag contains 8 red balls, 4 green, and 8 yellow balls. A ball is drawn at random from the bag and it is found not to be one of the red balls. What is the probability that it is a green ball? Solution Bag contains     8+8+4   =20    balls As this is not a red ball minus the number of red balls Total number of balls  8+4 =12 balls the probability that the ball which is got is green is = =0.333   23

24. SOME APPLICATIONS OF PROBABILITY There are several science and engineering applications of probability. Some of these applications are as follows: 1. Reliability Engineering Reliability theory is concerned with the duration of the useful life of components and systems of components. System failure times are unpredictable. Thus, the time until a system fails, which is referred to as the time to failure of the system, is usually modeled by a probabilistic function. 24

25. Some applications of probability ( contd ) 2. Quality Control Quality control deals with the inspection of finished products to ensure that they meet the desired requirements and specifications. One way to perform the quality control function is to physically test /inspect each product as it comes off the production line. A decision to declare the lot good or defective is thus based on the outcome of the test of the items of the sample. 25

26. BAYES’ THEOREM Bayes' theorem, named after 18th-century British mathematician Thomas Bayes, is a mathematical formula for determining conditional probability. The theorem provides a way to revise existing predictions or theories (update probabilities) given new or additional evidence. In finance, Bayes' theorem can be used to rate the risk of lending money to potential borrowers. 26

27. Bayes' theorem is well suited to and widely used in machine learning. Bayes' theorem is also called Bayes' Rule or Bayes' Law and is the foundation of the field of Bayesian statistics. This set of rules of probability allows one to update their predictions of events occurring based on new information that has been received, making for better and more dynamic estimates. 27

28. EXAMPLE 1.6 A student buys 1000 chips from supplier A, 2000 chips from supplier B, and 3000 chips from supplier C. He tested the chips and found that the probability that a chip is defective depends on the supplier from where it was bought. Specifically, given that a chip came from supplier A, the probability that it is defective is 0.05; given that a chip came from supplier B, the probability that it is defective is 0.10; and given that a chip came from supplier C, the probability that it is defective is 0.10 Given that a randomly selected chip is defective, what is the probability that it came from supplier A? 28

29. EXAMPLE 1.6 SOLUTION the probability that the randomly selected chip came from supplier A, given that it is defective, is given by P(D/A)=0.05, P(A)= = P(D/B) =0.10 , P(B)= = P(D/C) = 0.10, P(C)= =   29 P[A|D]= = P[A|D]= =0.0909 , 9.09%  

30. Activity In example 1.6 given that a randomly selected chip is defective, what is the probability that it came from supplier C? 30