Basic Factory Dynamics
Physics should be explained as simply as possible,
but no simpler.
Albert Einstein
Wallace J. Hopp, Mark L. Spearman, 1996, 2000
http:/factory-physics.com
1
HAL Case
Large Pan
NTHU MATH 2820, 2008
Solution to Homework 10
made by
NTHU MATH 2820, 2008
Solution to Homework 10
made by
NTHU MATH 2820, 2008
Solution to Homework 10
made by
NTHU MATH 2820, 2008
Solution to Homew
NTHU MATH 2820, 2008
Solution to Homework 9
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NTHU MATH 2820, 2008
Solution to Homework 9
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NTHU MATH 2820, 2008
Solution to Homework 9
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NTHU MATH 2820, 2008
Solution to Homework
NTHU MATH 2820, 2008
Solution to Homework 7
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NTHU MATH 2820, 2008
Solution to Homework 7
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NTHU MATH 2820, 2008
Solution to Homework 7
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NTHU MATH 2820, 2008
Solution to Homework 6
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NTHU MATH 2820, 2008
Solution to Homework 6
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NTHU MATH 2820, 2008
Solution to Homework 6
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NTHU MATH 2820, 2008
Solution to Homework
NTHU MATH 2820, 2008
Solution to Homework 4
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NTHU MATH 2820, 2008
Solution to Homework 4
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NTHU MATH 2820, 2008
Solution to Homework 4
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NTHU MATH 2820, 2008
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NTHU MATH 2820, 2008
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NTHU MATH 2820, 2008
3-70 Fx (x, y) = Pr ( X (1) x, X (n) y )
Solution to Homework 2
= Pr ( cfw_X (1) x cfw_X (n) y )
A
B
= Pr ( cfw_X (n) y ) Pr ( cfw_X (1) > x cfw_X (n) y ) ~ = Pr ( cfw_X1 , X 2 ,
NTHU MATH 2820, 2008
Solution to Homework 1
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NTHU MATH 2820, 2008
Solution to Homework 1
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NTHU MATH 2820, 2008
Solution to Homework 1
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NTHU MATH 2820, 2008
Solution to Homework
Expectation
p. 7-1
Recall. Expectation for univariate random variable. Theorem. For random variables X=(X1, . , Xn) with joint pmf pX/pdf fX, the expectation of a univariate random variable Y, where
Method of probability density function Theorem. Let X=(X1, ., Xn) be continuous random variables with the joint pdf fX. Let
p. 6-31
Y=(Y1, ., Yn)= g(X), where g is 1-to-1, so that its inverse exists a
Proof. Let Ai (y) = cfw_x : gi (x) y, i=1, ., n, then FY (y1 , . . . , yn ) = P (Y1 y1 , . . . , Yn yn ) = P (X1 A1 (y1 ), . . . , Xn An (yn ) = P (X1 A1 (y1 ) P (Xn An (yn ) = P (Y1 y1 ) P (Yn yn ) =
n n1,n m
=
n! n1 !n m !
p. 6-11
ways.
Example: MISSISSIPPI 11 4,1,2,4 =
11! 4!1!2!4! .
Example (Die Rolling). Q: If a balanced (6-sided) die is rolled 12 times, P(each face appears twice)=? Sample spa
Jointly Distributed Random Variables
Recall. In Chapters 4 and 5, focus on univariate random variable. However, often a single experiment will have more than one random variable which is of interest.
A special case of the gamma distribution occurs when =n/2 and =1/2 for some positive integer n. This is known as the Chi-squared distribution with n degrees of freedom (Chapter 6) Summary for X ~ Gamm
Example (Uniform Distributions). If 1 , if < x , - fX (x) = 0, otherwise, then
p. 5-11
Some properties of expectation Expectation of Transformation. If Y=g(X), then R R E(Y ) = - y fY (y) dy = - g(x)
Continuous Random Variables
p. 5-1
Recall: For discrete random variables, only a finite or countably infinite number of possible values with positive probability. Often, there is interest in random v
p. 4-31
Note: For X~binomial(n, p), where (i) n large; (ii) p small, distribution of X Poisson(np) E(X) np mean of the Poisson Var(X) np(1p) variance of the Poisson
Poisson Process Example: (1) # of
proof.
p. 4-21
Summary for X ~ Binomial(n, p) Range: X = cfw_0, 1, 2, ., n n x n-x , for x X Pmf: fX (x) = x p (1 - p) Parameters: ncfw_1, 2, 3, . and 0p 1 Mean: E(X)=np Variance: Var(X)=np(1p) Geomet
p. 4-11
Expectation (Mean) and Variance
Q: We often characterize a person by his/her height, weight, hair color, . How can we "roughly" characterize a distribution? Definition: If X is a discrete r.v
Random Variables
p. 4-1
A Motivating Example Experiment: Sample k students without replacement from the population of all n students (labeled as 1, 2, ., n, respectively) in our class. = cfw_all comb
Some Notes. The mgf is a function of the variable t. The mgf may only exist for some particular values of t. Example. If X is a discrete r.v. taking on values xi with probability pi, i=1, 2, 3, ., the
NTHU MATH 2810
Midterm Examination
Oct 30, 2007
Note. There are 7 problems in total. To ensure consideration for partial scores, write down intermediate steps where necessary.
1. (a) (3pts) Two years