Introduction to Probability
Uncertainty/Randomness (/) in our life Many events are random in that their result is unknowable before the event happens. Will it rain tomorrow? How many wins can achieve this season? What numbers will I roll on two dice? Q:
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 combinations = cfw_i1, ., ik: 1i1<ikn A probability measure
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. with pmf fX and range X , then the expectation (or ca
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) Geometric and Negative Binomial Distributions Experiment: A b
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 earthquakes occurring during some fixed time span (2) #
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 variables that can take (at least theoretically) on an u
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) fX (x) dx,
provided that the integral converges absolu
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 ~ Gamma(, ) -1 -x Pdf: x e , if x 0, () f(x) = 0, if x < 0. C
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.
p. 6-1
P
X1 X2 Xn
R R R
Definition. Given a sampl
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 space of rolling the die once (basic experiment): 0 = cfw_
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 ) = FY1 (y1 ) FYn (yn ). Theorem. X=(X1, ., Xn) are indepe
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 and is denoted by x=g1(y) = w(y) = (w1(y), w2(y), ., wn(
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 Y=g(X1, . , Xn), g:RnR1, is E(Y ) =
yY
y pY (y)
g(x1 ,
Proof. n (n - 1)S 2 = i=1 [(Xi - ) - (X n - )]2 n n 2 2 = + i=1 (Xi - ) i=1 (X n - ) n - 2(X n - ) [ i=1 (Xi - )] n 2 = + n(X n - )2 - 2n(X n - )2 i=1 (Xi - ) n 2 = - n(X n - )2 . i=1 (Xi - ) Therefore,
p. 7-11
Note. The previous three corollaries also ho
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, ., then If X ~ Poisson(), then for <t<,
MX (t) = = e
p. 7-21
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 ago, Santa Claus was distributing 10 identical candies
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 have values between 0 and 1. (c) False. It must be a one-
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: P(A)=? (What would you expect when n is large?) By symm
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 horses 1, 2, ., 7: = cfw_all 7! Permutations of (1, 2,
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 coins in order (assuming all choices are equally likely at
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 information about where you are staying P=? if you are stayin
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 )
dxdy
0 y L d y
dxdy d L
d 2 L
38. 1 2 Y 3 4 5 (a) P( X
NTHU MATH 2810, 2011
Solution to Homework 2
Jointly made by
NTHU MATH 2810, 2011
Solution to Homework 2
Jointly made by
NTHU MATH 2810, 2011
Solution to Homework 2
Jointly made by
NTHU MATH 2810, 2011
Solution to Homework 3
Jointly made by
NTHU MATH 2810, 2011
Solution to Homework 3
Jointly made by
NTHU MATH 2810, 2011
Solution to Homework 3
Jointly made by
NTHU MATH 2810, 2011
Solution to Homework 3
Jointly made by
NTHU MATH 2810, 2011
Solution to Homework 4
Jointly made by
NTHU MATH 2810, 2011
Solution to Homework 4
Jointly made by
NTHU MATH 2810, 2011
Solution to Homework 4
Jointly made by
NTHU MATH 2810, 2011
Solution to Homework 4
Jointly made by
NTHU MATH 2810, 2011
Solution to Homework 5
Jointly made by
NTHU MATH 2810, 2011
Solution to Homework 5
Jointly made by
NTHU MATH 2810, 2011
Solution to Homework 5
Jointly made by
NTHU MATH 2810, 2011
Solution to Homework 5
Jointly made by
NTHU MATH 2810, 2011
Solution to Homework 6
Jointly made by
NTHU MATH 2810, 2011
Solution to Homework 6
Jointly made by
NTHU MATH 2810, 2011
Solution to Homework 6
Jointly made by