Topic_ImportantProbabilityDistributions

# Topic_ImportantProbabilityDistributions - ECON634...

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ECON634: Econometrics and Business Statistics Topic 4 Some Important Probability Distributions 1 ECON634 Shuping Shi

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Lecturer Lecturer: Dr Shuping Shi - Email: [email protected] - Phone: 9850 8501 - Office: Room 441, E4A Building - Monday: 12:00 2:00pm (W5C220) - Consultation: Monday 3:00-4:00pm during teaching weeks ECON634 Shuping Shi 2
Probability Distributions Discrete probability distributions Binomial distribution Continuous probability distributions Normal distribution Chi-square distribution F-distribution T-distribution ECON634 Shuping Shi 3

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The Binomial Distribution A Bernoulli process has the following properties: There are two possible outcomes, which we call success and failure. The probability of a success is p , the probability of failure is (1- p ). If a fixed number, n , of Bernoulli trials are undertaken, the random variable representing the number of successes in the n trials has a binomial distribution . The trials are independent that is, the result of one trial does not affect the result of any other trials. ECON634 Shuping Shi 4
The Binomial Distribution Use notation X~Bin(n,p) Examples: Flip a coin 10 times, X = number of heads. Then X is a binomial random variable, X~Bin(n,p) n=10, p=0.5 Do a survey of 1000 people, X = number of people who think current PM is doing a good job. X is a binomial random variable, n=1000, p=? ECON634 Shuping Shi 5

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Binomial Random Variable The number of successes in n trials, denoted by X. Possible values : 0, 1, 2, …, n Discrete random variable Formula to calculate probabilities: ECON634 Shuping Shi 6 x n x p 1 p ! x n ! x ! n x X P 1 2 ... 2 n 1 - n n n! : Note
Example Example: A student sitting an econometric quiz decides to answer each of the 10 multiple choice questions entirely by chance. Each question has 5 options, only one of which is correct. Let X be the number of questions the student answers correctly. ECON634 Shuping Shi 7

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Is this a binominal experiment? conditions: There is a fixed finite number of trials ( n=10 ). An answer can be either correct or incorrect. Each answer is independent of the others. The probability of a correct answer (P(success)=.20) does not change from question to question. Then, X~Bin(n=10, p=0.2) ECON634 Shuping Shi 8
What is the probability the student gets no answers correct? ECON634 Shuping Shi 9 dp) 5 (to 10737 . 0 8 . 0 1 1 2 . 0 1 2 . 0 ! 0 10 ! 0 ! 10 0 X P 10 0 10 0 x n x p 1 p ! x n ! x ! n x X P

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What is the probability that the student passes (i.e. gets 5 or more correct)? ECON634 Shuping Shi 10 calculate! to formulas of lot a 10 X P 9 X P 8 X P 7 X P 6 X P 5 X P 5 X P