Probability & probability distribution

Probability & Probability Distribution Unit IV



Transcript of Probability & probability distribution

  • 1. Unit IV

2. Probability is the likelihood or chance that a particular event will or will not occur; The theory of probability provides a quantitative measure of uncertainty of occurrence of different events resulting from a random experiment, in terms of quantitative measures ranging from 0 to 1; 3. Experiment: it is a process which produces outcomes; Example, tossing a coin is an experiment; an interview to gauge the job satisfaction levels of the employees in an organization is an experiment; Event: it is the outcome of an experiment; Example, if the experiment is to toss a fair coin, an event can be obtaining a head; if an event has a single possible outcome, then it is a simple (or elementary) event; a subset of outcomes corresponding to a specific event is called an event space. 4. Independent & Dependent Events: two events are said to be independent, if the occurrence or non-occurrence of one is not affected by the occurrence or non-occurrence of the other; vice versa Mutually Exclusive Events: two or more events are said to be mutually exclusive if the occurrence of one implies that the other cannot occur; if X and Y are mutually exclusive, then P(XY)=0 Sample Space: denoted by S; it is the set of all possible outcomes in an experiment; 5. Classical/ Prior Approach; Relative sequence/Empirical Approach;& Subjective/Intuitive/Judgmental Approach. 6. This approach happens to be the earliest; This school of thought assumes that all the possible outcomes of an experiment are mutually exclusive & equally likely; If there are a possible outcomes favorable to the occurrence of Event E, & b possible outcomes unfavorable to the occurrence of Event E & all these possible outcomes are equally likely & mutually exclusive, then the probability that the event E will occur, denoted by P(E), is P(E)= Number of outcomes favorable to occurrence of E Total number of outcomes 7. This approach has two characteristics: a. The subjects refers to fair coins, fair decks of cards; but if the coin is unbalanced or there is a loaded dice, this approach would offer nothing but confusion; b. In order to determine probabilities, no coins had to be tossed, no cards shuffled, i.e. no experimental data were required to be collected; 8. This method uses the relative frequencies of past occurrences as the basis of computing present probability; hence it is based on experiments conducted in the past; If an Event E has occurred r number of times in a series of n independent trials;, all under uniform conditions, then the ratio of r gives the probability of Event E provided n is sufficiently large: P(E)= r = favorable trials n total of trials 9. This approach is based on the intuition of an individual; This is not a scientific approach; It is based on accumulation of knowledge, understanding and experience of an individual; 10. For any event probability lies between 0 & 1; It is represented in percentages, ratios, fractions; Each event has a complementary event i.e. P(E1) + P(E1) =1 11. Marginal Probability; Union Probability; Joint Probability; Conditional Probability. 12. It is the first type of probability; A marginal or unconditional probability is the simple probability of the occurrence of an event; Denoted by P(E) where E is some event; P(E)= Number of outcomes favorable to occurrence of E Total number of outcomes 13. Second type of probability; If E1 & E2 are two Events, then Union probability is denoted by P(E1 U E2 ); It is the probability that Event E1 will occur or that Event E2 will occur or both Event E1 & Event E2 will occur; For example, union probability is the probability that a person either owns a Maruti 800 or Maruti Zen. For qualifying to be part of the union, a person has to have atleast one of these cars 14. It is the third type of probability; If E1 & E2 are two Events, then Joint probability is denoted by P(E1E2 ); It is the probability of the occurrence of Event E1 and Event E2; For example, it is the probability that a persons owns both a Maruti 800 & Maruti Zen; for joint probability, owning a single car is not sufficient; 15. It is the fourth type of probability; Conditional Probability of two Events E1 & E2 is generally denoted by P(E1/E2); It is probability of the occurrence of E1 given that E2 has already occurred; Conditional probability is the probability that a person owns a Maruti 800 given that he already has a Maruti Zen; 16. Used to estimate union probability; If there are two Events E1 & E2, then the general rule of addition is given by: P(E1 or E2) = P(E1) + P(E2) P (E1 & E2); P(E1 U E2) = P(E1) + P(E2) P (E1E2); Special Rule of addition for mutually exclusive: P(E1 or E2) = P(E1) + P(E2); P(E1 U E2) = P(E1) + P(E2); 17. Used to estimate joint probability and also conditional probability; If there are two Events E1 & E2, then the general rule of multiplication is given by: P(E1 & E2) = P(E1) . P(E2 /E1); P(E1 E2) = P(E1) . P(E2 /E1) ; Special Rule of multiplication for independent events: P(E1 & E2) = P(E1) . P(E2); P(E1 E2) = P(E1) . P(E2); 18. Bayes theorem was developed by Thomas Bayes. In fact, Bayes theorem is an extended use of the concept of conditional probability; The law of conditional probability is given by: P(E1/E2) = P(E1 E2) = P(E1) . P(E2 /E1) P(E2) P(E2) 19. A random variable is a variable which contains the outcome of a chance experiment; for example, in an experiment to measure the number of customers who arrive in a shop during a time interval of 2 minutes; the possible outcome may vary from 0 to n customers; these outcomes (0,1,2,3,4,n)are the values of the random variable. These random variables are called discrete random variables 20. In other words , a random variable which assumes either a finite number of values or a countable infinite number of possible values is termed as Discrete Random variable On the other hand, random variables that assumes any numerical value in an interval or can take values at every point in a given interval is called continuous random variable. For example, temperatures recorded for a particular city can assume any number like 32O F, 32.5O F 35.8O F Experiment outcomes which are based on measurement scale such as time, distance, weight & temperature can be explained by Continuous Random variable 21. Most commonly used & widely known distribution among all discrete distributions. It is a sequence of repeated trials, called Bernoulli Process which is characterized by: 1. Only two mutually exclusive outcomes are possible;( one is referred to as success & the other as failure) 2. The outcomes in a series of trials/observation constitute independent events; 3. Probability of success (p) or failure (q) is constant over a number of trials; 4. The number of events is discrete & can be represented by integers(0,1,2,3,4,onwards) 22. P(X)= nCxpxqn-x where n= total number of trials x = Designated value p= probability of success q= probability of failure nCx= n!___ x!(n-x)! 23. It is named after the famous French Mathematician Simeon Poisson; It is also a discrete distribution; but there are a few differences between Binomial & Poisson distributions. For a given number of trials the binomial distribution describes a distribution of two possible outcomes: either success or failure whereas Poisson focuses on the number of discrete occurrences over an interval. It is widely used in the field of managerial decision making; widely used in queuing models 24. The event occur in a continuum of time & at a randomly selected point & event either occurs or doesnt occur; Whether the event occur or doesnt occur at a point, it is independent of the previous point where the event may have occurred or not; The probability of occurrence of events remains same/constant over the whole period or throughout the continuum; 25. P(x/)= x e-x! (greek letter lambda) =mean/average e (constant)= 2.71826 x is a random variable(designated number) 26. It is the most commonly used distribution among all probability distributions; It has a wide range of practical application example, where the random variables are human characteristics such as height, weight, speed, IQ scores; Normal distribution was invented in the 18th century; 27. The curve of normal distribution is symmetrical/ mesokurtic; The mean, median & mode are identical; The two tail of normal curve asymptotic; Curve is unimodal; The total area under normal distribution is 100% & the distribution is as follows: +1 = 68% +2 =97% +3 = 99.7% Z= x-