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

SLIDE 1

1

• G. Ahmadi

ME 529 - Stochastics

• G. Ahmadi

ME 529 - Stochastics

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

• G. Ahmadi

ME 529 - Stochastics

Series of Independent Experiments Series of Independent Experiments

( )

p a P =

( )

q a P =

1 = +q p

( )

k n k n

q p k n k P

−

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = Probability that Probability that event event a a occurs k

• ccurs k

times in n trials: times in n trials:

( )

k n k

• rder

Specific n

q p k P

−

=

Probability that event Probability that event a a

• ccurs k times in specific
• ccurs k times in specific
• rder in n trials:
• rder in n trials:
• G. Ahmadi

ME 529 - Stochastics

( )

k n k n

q p k n k P

−

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = Pn(k) k

4 2 8 6

max

k

SLIDE 2

2

• G. Ahmadi

ME 529 - Stochastics

Most Likely Number of Success Most Likely Number of Success

( ) ( ) ( ) ( )

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ = + ≠ + + = + ≤ = − = Integer P n if Integer P n if P n k P n Integer Greatest k k and k k k 1 1 1 1 1

1 1 1 1 1 max

( ) ∑

= −

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ≤ ≤

2 1

2 1 k k k k n kq

p k n k k k P

1 = + = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛

∑

= − n n k k n k

) q p ( q p k n

• G. Ahmadi

ME 529 - Stochastics

( )

( )

npq np k k n k n

e npq q p k n k P

2

2

2 1

− − − ≈

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = π

( )

( )

∑ ∑

= = − − = = − ≈

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ≤ ≤

2 1 2 2 1

2 2 1

2 1

k k k k npq np k k k k k k n k n

e npq q p k n k k k P π

DeMoivre DeMoivre – – Laplace Laplace Theorem Theorem

For n For n→∞ →∞, , npq npq>>1 >>1 k~(npq) k~(npq)1/2

1/2 nbh

nbh of

• f

np np Stirling Stirling Formula Formula

n e n ! n

n n

π 2

−

=

• G. Ahmadi

ME 529 - Stochastics

( )

npq np k erf npq np k erf k k k P

n

− − − ≈ ≤ ≤

1 2 2 1

Approximate Evaluation Approximate Evaluation

( )

( )

dx e npq k k k P

k k k k npq np x n

∫

= = − −

≈ ≤ ≤

2 1 2

2 2 1

2 1 π

( )

∫

−

=

x y

dy e x erf

2

2

2 1 π

( )

2 1 = ∞ erf

• G. Ahmadi

ME 529 - Stochastics

( )

2

2

2 1

x

e x g

−

= π ( )

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ≈ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

−

npq np k g npq q p k n k P

k n k n

1

( ) ( )

∫ ∫

∞ − − ∞ −

= =

x y x

dy e dy y g x G

2

2

2 1 π ( )

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = ≤ ≤ npq np k G npq np k G k k k P

n 1 2 2 1

g, G g, G

SLIDE 3

3

• G. Ahmadi

ME 529 - Stochastics

( )

r

k r k k r r n

p ... p p ! k !... k ! k ! n k ,..., k , k P

2 1

2 1 2 1 2 1

= DeMoivre DeMoivre – – Laplace Laplace

= ∩

j i

a a

( )

r r

p a P =

For For With With

( ) ( ) ( ) ( )

r r r r r r n

p ... p n np np k ... np np k exp k ,..., k , k P

1 1 2 1 2 1 1 2 1

2 2 1

−

⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − + + − − ≈ π

• G. Ahmadi

ME 529 - Stochastics

For large n, small p with For large n, small p with np np being finite being finite

( ) ( )

! k a e ! k np e q p k n k P

k a k nP k n k n − − −

= ≈ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

( ) ( )

∑

= −

≈ ≤ ≤

2 1

2 1 k k k k nP n

! k np e k k k P

• G. Ahmadi

ME 529 - Stochastics

We place at random n points in the interval We place at random n points in the interval (0,T). Let t (0,T). Let t2

2 –

– t t1

1 =

= t ta

• a. The probability of

. The probability of finding k points in finding k points in t ta

a is

is

( )

k n k a

q p k n t in s int Po k P

−

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = T t p

a

=

Poisson Poisson

( ) ( )

! k t e t in s int Po k P

k a t a

a λ

λ −

≈ T n = λ

• G. Ahmadi

ME 529 - Stochastics

Concluding Remarks Concluding Remarks

• 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