1.21.10 Lec 4

1.21.10 Lec 4 - Lecture 4 Binomial and Poisson...

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1 Lecture 4 Binomial and Poisson Distributions 1 Lecture 4 Outline 1. Counting Techniques ¾ Factorial ¾ Permutations ¾ Combinations 2. Binomial Distribution 3. Binomial Example (n=3) 4. Graphing a Binomial Probability Distribution 5. Poisson Distribution 6. Poisson Approximation of the Binomial Distribution 7. Mean and Variance of a Distribution 2

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2 1. Counting Techniques 1.1 Factorials 3 objects Notation: “n factorial” = n! = number of possible arrangements of n objects n! = n(n-1)(n-2)(n-3) … (2)(1) •E x a m p l e : 3 ! = 3 × 2 × 1 = 6 arrangements ABC ACB BAC BCA CAB arrangements x a m p l e : 4 ! = 4 × 3 × 2 × 1 = 24 arrangements •0 ! = 1 3 CBA 1. Counting Techniques (cont’d) 1.2 Permutations 3 objects • Permutation = ordered arrangement of n objects taken r at a time n P r = n!/(n-r)! AB BA AC 3 objects taken 2; order matters Example: 3 P 2 = 3!/(3-2)! = 3!/1!= 3(2) = 6 x a m p l e : 4 P 2 = 4!/(4-2)! = 4!/2!= 4(3) = 12 4 CA BC CB
3 1. Counting Techniques (cont’d) 1.3 Combinations Combination = an arrangement of n objects take r at a time without regard to order () ! " " !-! nr n n C n choose r r rnr ⎛⎞ == = ⎜⎟ ⎝⎠ AB/BA AC/CA 3 objects taken 2 order unimportant •E x a m p l e : 3 C 2 = 3!/[2!(3-2)!] = 3!/(2!1!)= 6/2 = 3 x a m p l e : 4 C 2 = 4!/[2!(4-2)!] = 4!/(2!2!)= 12/2 = 6 5 BC/CB 2.Binomial Distribution Assumptions: 1. Bernoulli trial model a) The study or experiment consists of n smaller experiments (trials) each of which has only two possible outcomes, e.g. dead, alive success, failure diseased, not diseased b) The outcomes of the trials are independent c) The probabilities of the outcomes of the trials remain the same from trial to trial 6

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4 2. Binomial Distribution (cont’d) 2. The probability of obtaining x “successes” in n Bernoulli trials is Bernoulli trials is where p = probability of a “success” in each independent trial and q = 1-p = probability of a “failure” () n x nx PX x pq x ⎛⎞ == ⎜⎟ ⎝⎠ Note 1 : Tables, calculators, and computers are available for calculating these probabilities! Note 2 : The notation P(X=x) represents Pr (X=x) 7 3.1 Binomial Example n=3 • Let n = 3, p = probability of success” then X (the number of “successes") can equal 0, 1, 2 or 3 8
5 3.1 Binomial Example n=3 (cont’d) •P ( X 1) = P(X=0) + P(X=1)

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1.21.10 Lec 4 - Lecture 4 Binomial and Poisson...

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