Probability & probability distribution


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Probability & probability distribution

  1. 1. Unit IV
  2. 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. 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. 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(X∏Y)=0  Sample Space: denoted by S; it is the set of all possible outcomes in an experiment;
  5. 5. Classical/ Prior Approach; Relative sequence/Empirical Approach;& Subjective/Intuitive/Judgmental Approach.
  6. 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. 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. 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. 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. 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. 11. Marginal Probability; Union Probability; Joint Probability; Conditional Probability.
  12. 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. 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. 14. It is the third type of probability; If E1 & E2 are two Events, then Joint probability is denoted by P(E1∏E2 ); 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. 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. 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 (E1∏E2); Special Rule of addition for mutually exclusive: P(E1 or E2) = P(E1) + P(E2); P(E1 U E2) = P(E1) + P(E2);
  17. 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. 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. 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. 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. 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. 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. 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. 24. The event occur in a continuum of time & at a randomly selected point & event either occurs or doesn’t occur; Whether the event occur or doesn’t 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. 25.  P(x/)= x e- x! (greek letter lambda) =mean/average e (constant)= 2.71826 x is a random variable(designated number)
  26. 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. 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: µ+1σ = 68% µ+2σ =97% µ+3σ = 99.7% Z= x- µ σ