Understanding the Binomial Distribution and Its Applications
Explore the concept of binomial distribution, a probability distribution that models the number of successes in a fixed number of independent trials. Learn about its assumptions, mean, variance, and standard deviation, as well as practical applications in various fields like clinical trials, quality control, and more.
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Presentation Transcript
APPLICATION OF BINOMIAL APPLICATION OF BINOMIAL DISTRIBUTION DISTRIBUTION
WHAT IS BINOMIAL DISTRIBUTION ? IF X IS A DISCRETE RANDOM VARIABLE WITH PROBABILITY MASS FUNCTION. WHERE X=0,1,2,3........N&Q=1-P, THEN X IS A BINOMIAL VARIATE AND THE DISTRIBUTION OF X IS CALLED BINOMIAL DISTRIBUTION. THE WORD BINOMIAL LITERALLY MEANS TWO NUMBERS. A BINOMIAL DISTRIBUTION FOR A RANDOM VARIABLE X (KNOWN AS BINOMIAL VARIATE) IS ONE IN WHICH THERE ARE ONLY TWO POSSIBLE OUTCOMES, SUCCESS AND FAILURE, FOR A FINITE NUMBER OF TRIALS.
NOTE THAT HOWEVER WE DEFINE SUCCESS AND FAILURE, THE TWO EVENTS MUST BE MUTUALLY EXCLUSIVE AND COMPLEMENTARY; THAT IS, THEY CANNOT OCCUR AT THE SAME TIME (MUTUALLY EXCLUSIVE), AND THE SUM OF THEIR PROBABILITIES IS 100% (COMPLEMENTARY). WHERE P Q SHOULD BE GREATER THAN 1. X IS A DISCRETE RANDOM VARIABLE WHICH MAY TAKE ON ONLY COUNTABLE NUMBER OF DISTINCT VALUES SUCH AS 1,2,3,4..... P+Q=1 BECAUSE PROBABILITY CANNOT BE MORE THAN 1.
ASSUMPTIONS FOR BINOMIAL DISTRIBUTION FOR EACH TRIAL THERE ARE ONLY TWO POSSIBLE OUTCOMES ON EACH TRIAL, S (SUCCESS) & F (FAILURE). THE NUMBER OF TRIALS N IS FINITE. FOR EACH TRIAL, THE TWO OUTCOMES ARE MUTUALLY EXCLUSIVE. P(S) = P IS CONSTANT. P(F) = Q = 1-P. THE TRIALS ARE INDEPENDENT, THE OUTCOME OF A TRIAL IS NOT AFFECTED BY THE OUTCOME OF ANY OTHER TRIAL. THE PROBABILITY OF SUCCESS, P, IS CONSTANT FROM TRIAL TO TRIAL.
MEAN, VARIANCE, AND STANDARD DEVIATION FOR THE BINOMIAL DISTRIBUTION IF X HAS A BINOMIAIS DISTRIBUTION WITH N TRIALS AND PROBABILITY OF SUCCESS P ON EACH TRIAL, THEN: THE MEAN OF X IS : THE VARIANCE OF X IS : THE STANDARD DEVIATION OF X IS :
APPLICATIONS FOR BINOMIAL DISTRIBUTIONS BINOMIAL DISTRIBUTIONS DESCRIBE THE POSSIBLE NUMBER OF TIMES THAT A PARTICULAR EVENT WILL OCCUR IN A SEQUENCE OF OBSERVATIONS. THEY ARE USED WHEN WE WANT TO KNOW ABOUT THE OCCURRENCE OF AN EVENT, NOT ITS MAGNITUDE. EXAMPLES: 1. IN A CLINICAL TRIAL, A PATIENT S CONDITION MAY IMPROVE OR NOT. WE STUDY THE NUMBER OF PATIENTS WHO IMPROVED, NOT HOW MUCH TTER FEEL. 2. IS A PERSON AMBITIOUS OR NOT? THE BINOMIAL DISTRIBUTION DESCRIBES THE NUMBER OF AMBITIOUS PERSONS, NOT HOW AMBITIOUS THEY ARE. 3. IN QUALITY CONTROL WE ASSESS THE NUMBER OF DEFECTIVE ITEMS IN A LOT OF GOODS, IRRESPECTIVE OF THE TYPE OF DEFECT.
AREAS OF APPLICATION COMMON USES OF BINOMIAL DISTRIBUTIONS IN BUSINESS INCLUDE QUALITY CONTROL. INDUSTRIAL ENGINEERS ARE INTERESTED IN THE PROPORTION OF DEFECTIVES ALSO USED EXTENSIVELY FOR MEDICAL (SURVIVE, DIE) IT IS ALSO USED IN MILITARY APPLICATIONS (HIT, MISS).
