( ) n ( ) = ( ) k n k + q = P k p q k = = p - PowerPoint PPT Presentation

Outline Outline � Repeated Trials � Repeated Trials � Most Likely Number of Success � Most Likely Number of Success � Asymptotic Theorems � Asymptotic Theorems � Gaussian Functions � Gaussian Functions � Generalized Bernoulli Trials � Generalized Bernoulli Trials � Poisson Theorem � Poisson Theorem � Random Poisson Points � Random Poisson Points ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi Series of Independent Experiments Series of Independent Experiments ⎛ ⎞ ( ) n ( ) = ⎜ ⎟ − ( ) k n k + q = P k ⎜ ⎟ p q k = = p 1 n ⎝ ⎠ k P a p P a q max P n (k) ( ) Probability that event a Probability that event a = − k n k P k p q occurs k times in specific occurs k times in specific n Specific order order in n trials: order in n trials: Probability that Probability that ⎛ ⎞ n ( ) = ⎜ ⎟ − k n k event a event a occurs k occurs k P k p q ⎜ ⎟ 2 4 6 8 n ⎝ ⎠ k times in n trials: times in n trials: k ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 1

DeMoivre – – Laplace Laplace Theorem Theorem DeMoivre Most Likely Number of Success Most Likely Number of Success For n →∞ →∞ , ( ) For n , npq npq>>1 >>1 − 2 ⎛ ⎞ k np − ≈ − ( ) ( ) ( ) n 1 = ≤ + + ≠ ⎧ ⎫ k k Greatest Integer n 1 P if n P Integer 1/2 nbh = ⎜ ⎟ 1 k n k k~(npq) 1/2 2 npq k~(npq) nbh of of P k p q e = 1 1 ⎜ ⎟ ⎨ ⎬ k ( ) ( ) − = + + = n π max ⎝ ⎠ ⎩ ⎭ k k and k 1 k n 1 P if n 1 P Integer 2 npq np np 1 1 1 = − π Stirling Formula Formula Stirling n n ⎛ ⎞ ) ∑ k n ! n e 2 n n ( 2 ≤ ≤ = ⎜ ⎟ − k q n k P k k k p ⎜ ⎟ 1 2 ⎝ ⎠ k = k k 1 ( ) − 2 = ⎛ ⎞ = k np k k k k − n − ≈ ( ) ∑ 1 ∑ 2 2 ≤ ≤ = ⎜ ⎟ k n k 2 npq ⎛ ⎞ P k k k p q e ⎜ ⎟ n n ∑ ⎜ ⎟ − = + = π k n k n n 1 2 ⎜ ⎟ p q ( p q ) 1 ⎝ ⎠ k 2 npq ⎝ ⎠ = = k k k k k = k 0 1 1 ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi ( ) = − 2 k k x np − ( ) 1 2 2 2 x y ∫ ≤ ≤ ≈ − − ( ) 1 ( ) ( ) 1 2 npq x x ∫ ∫ P k k k e dx = = = g x e 2 G x g y dy e 2 dy π n 1 2 π π 2 npq − ∞ − ∞ 2 2 = k k 1 Approximate Evaluation Approximate Evaluation ⎛ ⎞ ⎛ ⎞ − n ( ) 1 k np ⎜ ⎟ = ⎜ ⎟ − ≈ k n k P k ⎜ ⎟ p q g ⎜ ⎟ − − n ( ) k np k np ⎝ ⎠ k npq ⎝ npq ⎠ ≤ ≤ ≈ − 2 1 P k k k erf erf n 1 2 npq npq g, G g, G ⎛ ⎞ ⎛ ⎞ − − ( ) k np k np ⎜ ⎟ ⎜ ⎟ ≤ ≤ = − 2 y 2 1 ( ) P k k k G G ( ) 1 − 1 ⎜ ⎟ ⎜ ⎟ ∫ x = ∞ = n 1 2 erf x e 2 dy erf ⎝ ⎠ ⎝ ⎠ npq npq π 0 2 2 ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 2

( ) For large n, small p with np For large n, small p with np being finite being finite ∩ = = For a a 0 With P a p For With i j r r ( ) ⎛ ⎞ k k n ( ) np a ( ) n ! = ⎜ ⎟ − ≈ − = − = k n k nP a k k k P k p q e e P k , k ,..., k p p ... p ⎜ ⎟ 1 2 r n 1 2 r 1 2 r n k ! k !... k ! ⎝ ⎠ k k ! k ! 1 2 r DeMoivre – – Laplace Laplace DeMoivre ( ) k k ( ) np ∑ 2 ≤ ≤ ≈ − nP P k k k e ( ) ( ) ⎧ ⎫ ⎡ ⎤ ⎪ − 2 − 2 ⎪ 1 k np k np n 1 2 − + + ⎨ ⎢ 1 1 r r ⎥ ⎬ k ! exp ... = ⎪ ⎪ k k 2 ⎣ np np ⎦ ⎩ ⎭ ( ) 1 1 r ≈ P k , k ,..., k ( ) n 1 2 r π − r 1 2 n p ... p 1 r ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi Concluding Remarks We place at random n points in the interval Concluding Remarks We place at random n points in the interval (0,T). Let t 2 – t t 1 = t t a . The probability of (0,T). Let t 2 – 1 = a . The probability of � Repeated Trials � Repeated Trials finding k points in t t a is finding k points in a is � Most Likely Number of Success � Most Likely Number of Success ⎛ ⎞ n � Asymptotic Theorems ( ) � t Asymptotic Theorems = ⎜ ⎟ − = k n k a P k Po int s in t ⎜ ⎟ p q p a ⎝ ⎠ k � Gaussian Functions T � Gaussian Functions � Generalized Bernoulli Trials � Generalized Bernoulli Trials Poisson Poisson ( ) a λ � Poisson Theorem � k n ( ) Poisson Theorem t λ = ≈ − λ t a P k Po int s in t e a T � Random Poisson Points � k ! Random Poisson Points ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 3