n k n k q p k p q k p 1 n
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( ) n ( ) = ( ) k n k + q = P k p q k = = p - PowerPoint PPT Presentation

Dec 18, 2022 •205 likes •244 views

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

  1. 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

  2. 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

  3. ( ) 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

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