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
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
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. 4-31
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
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
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)
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
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.
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
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 ) =
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
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
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
NTHU MATH 2810
1. (16pts, 2pts for each) (a) True.
Final Examination Solution
Jan 8, 2008
(b) False. The values of a pdf can be larger than 1. However, the integration of a pdf over any region must ha
Example (The Matching Problem). Applications: (a) Taste Testing. (b) Gift Exchange. Let be all permutations = (i , ., i ) of 1, 2, ., n. 1 n Thus, # = n!. Let Aj = cfw_: ij = j and A = n Ai , i=1 Q:
Sample Space and Events
p. 2-1
Sample Space : the set of all possible outcomes in a random phenomenon. Examples: 1.Sex of a newborn child: = cfw_girl, boy 2.The order of finish in a race among the 7
Example (Gold Coins): The Story. Box 1 contains 2 silver coins. Box 2 contains 1 gold and 1 silver coin. Box 3 contains 2 gold coins. Experiment: (i) Select a box at random and, (ii) Examine the 2 coi
Conditional Probability
Q: Should the following probabilities be different? Event = 6 P=? in the beginning P=? in the middle P=? of the season of the season now Event = rain tomorrow P=? if no inform
NTHU MATH 2810, 2011
Solution to Homework 9
14. X: Y: Let D | X FD ( d ) P( D P (Y 2 L2 2 L2 Y| d) X P (| X Y dxdy
0 y L d y d L L
~ Unif (0, L ) ~ Unif (0, L ) Y | d) P( X Y X d)
d)
L m in( y d , L )
NTHU MATH 2810, 2011
Solution to Homework 2
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NTHU MATH 2810, 2011
Solution to Homework 2
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NTHU MATH 2810, 2011
Solution to Homework 2
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NTHU MATH 2810, 2011
Solution to Homework 3
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NTHU MATH 2810, 2011
Solution to Homework 3
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NTHU MATH 2810, 2011
Solution to Homework 3
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NTHU MATH 2810, 2
NTHU MATH 2810, 2011
Solution to Homework 4
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NTHU MATH 2810, 2011
Solution to Homework 4
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NTHU MATH 2810, 2011
Solution to Homework 4
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NTHU MATH 2810, 2
NTHU MATH 2810, 2011
Solution to Homework 5
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NTHU MATH 2810, 2011
Solution to Homework 5
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NTHU MATH 2810, 2011
Solution to Homework 5
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NTHU MATH 2810, 2
NTHU MATH 2810, 2011
Solution to Homework 6
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NTHU MATH 2810, 2011
Solution to Homework 6
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NTHU MATH 2810, 2011
Solution to Homework 6
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