continuing from previous lecture1.5
Examples for multinomial coecients Ex.8- Painting. A house has 12 rooms. We want to paint 4 yellow, 3 purple, and 5 red. In how many ways can this be done?
continuing from previous lecture1.5
Examples for multinomial co
Lecture 5- Outline and Examples (3.2 and 3.3 in Ross)
Conditional Probability continued
Multiplication rule Rule of average conditional probabilities partition of S
Bayes formula
Conditional Probability (Ross 3.2)
If P (B ) > 0, then the probability that
PSTAT 120A, Spring 2010, Midterm. April 30, 2010 - in class exam This exam consists of 4 questions. Answer all questions. Clear working must be shown to receive credit. You may answer the 4 main questions in any order, but please keep the subparts of the
Lecture 16- Outline and Examples
Continuous random variables (Ross Ch 5) -Uniform random varibale (Ross 5.3) -Normal (Gaussian) random variable (Ross 5.4)
Uniform random variables (Ross 5.3)
The uniform distribution on (a, b ). Write: Y U (a, b ). fY (y )
Lecture 14- Outline and Examples
Continuous random variables (Ross Ch 5) -Denition (Ross 5.1) -Expected value and variance(Ross 5.2) -Uniform random varibale (Ross 5.3)
Probability density functions
Example Experience has shown that while walking in a cer
Lecture 12- Outline and Examples
Discrete probability distributions -Negative Binomial (Ross 4.8.2) -Properties of cumulative distribution functions (Ross 4.10) -Review of Chapters 3 and 4
Negative Binomial Distribution. 4.8.2
Let Xr denote the number of
Lecture 11- Outline and Examples
Discrete probability distributions -Geometric (Ross 4.8.1) -Negative Binomial (Ross 4.8.2)
Geometric distribution
Example. Rolling a die. Problem 1. Roll a symmetric die until you have rolled a six. What is the chance that
Lecture 7- Outline and Examples
Discrete probability distributions -Poisson -Geometric -Negative Binomial
Poisson distribution ( Ross chapter 4)
Example. Defective in a sample. Problem. Suppose that over the long run a manufacturing process produces 1% de
Lecture 8- Outline and Examples
Examples of discrete random variables -Hypergeometric distribution -Bernoulli trials and Binomial distribution
Hypergeometric distribution
Example: Random sampling without replacing. Random sampling is a statistical techniq
Lecture 7- Outline and Examples
Discrete vs Continuous probability functions Random variables -denition -cumulative distribution function Discrete Random variables -denition -probability mass function -expected value -variance
Discrete case ( Ross chapter
Lecture 7- Outline and Examples (3.4 in Ross)
Independence - Two independent events -More than two independent events -Pairwise independent events Random variables
Independent Events (Ross 3.4)
Recall Multiplication rule: P (A B ) = P (A|B )P (B ) = P (B
Lecture 4- Outline and Examples (2.5 and 3.2 in Ross)
Sample spaces with equally likely outcomes Conditional Probability
Equally likely outcomes ( Ross 2.5)
If all outcomes in the sample space are equally likely, then for any event A, P (A) = |A|/|S |, i.
Lecture 3- Outline and Examples (2.3-2.5 in Ross)
Axioms of Probabiliy basic properties of probabilities
Applying basic properties of probabilities
Example. Rich and Famous. In a certain population, 10% of the people are rich, 5% are famous, and 3% are ri
Lecture 1: Combinatorial analysis. Text: Based on 1.1-1.5 Tool used in calculating probabilities Combinatorics: theory of counting. Basic facts from combinatorics: Multiplication principle/Basic Principle of counting: If one experiment has m outcomes and
Chapter 3. Conditional probability denition: P (A B ) . P (A|B ) = P (B ) Multiplication rule P (A B ) = P (B ) P (A|B ) = P (A) P (B |A). Denition of a partition of S . If A1 , A2 , . . . , An is a partition of S (a special case being A and Ac ) then rul
PSTAT 120A, Spring 2010, Homework Assignment 9. Due Wednesday June 2 in lecture. No late homework. Reading: Ross Sections 6.4, 6.5, 7.2 (p. 297-302), 7.4 and 8.2 Complete and clear working must be shown to receive credit. p.289-290 Problem 40 p. 290 Probl
PSTAT 120A Spring 2010 HW 9 Solutions
1. (Ross 6.40) (a) p(x, y ) is the joint PMF p(x, y ) = pX,Y (x, y ) = P (X = x, Y = y ) of discrete random variables X and Y . The conditional PMF of X given Y can be obtained via pX |Y (x | y ) = So we need the marg
PSTAT 120A, Spring 2010, Homework Assignment 8. Due Wednesday May 26 in lecture. No late homework. Reading: Ross Sections 6.1, 6.2 and 6.3 Complete and clear working must be shown to receive credit. p. 287 Problem 2 p. 287 Problem 9 p. 288 Problem 20 p. 2
PSTAT 120A, Spring 2010, Homework Assignment 7. Due Wednesday May 19 in lecture. No late homework. Reading: Ross Sections 5.4.1, 5.5, 5.6.1 and 5.7 Complete and clear working must be shown to receive credit. If X Exp(), nd E [X ] and Var(X ). p. 226 Probl
PSTAT 120A Spring 2010 HW 7 Solutions
1. X exp() E [X ] = =
0
xf (x)dx,
xex dx.
To evaluate this integral, we use integration by parts b x=b u(x)dv (x) = u(x)v (x) |x=a
a a
b
v (x)du(x).
For our purposes we set u(x) = x and dv (x) = ex dx. Then we get
PSTAT 120A, Spring 2010, Homework Assignment 6. Due Wednesday May 12 in lecture. No late homework. Reading: Ross Sections 5.3-5.4 Complete and clear working must be shown to receive credit. If X U (a, b), nd E [X ] and Var(X ). p. 225 Problem 11 p. 225 Pr
solution to hw6
1 , X U (a, b), that is, its pdf is: f (x) = Hence, E[X ] =
a
1 , if a x ba 0, otherwise.
b
b;
x
1 b+a dx = ; ba 2 1 b2 + ab + a2 = . ba 3
and E[X 2 ] =
a
b
x2
1 b3 a3 dx = ba 3
Therefore, V ar[X ] =
b2 + ab + a2 b+a 2 (b a)2 ( )= . 3 2 12
PSTAT 120A, Spring 2010, Homework Assignment 5. Due Wednesday May 5 in lecture. No late homework. Reading: Ross Sections 5.1-5.2 Give a detailed solution with an argument, not just a numerical answer, to receive full credit. A numerical nal answer is requ
PSTAT 120A Spring 2010 HW 5 Solutions
1. (Ross 5.3) For a function to be a valid pdf, it must be nonnegative everywhere, and integrate to 1. For the rst function, check the endpoints of the domain (0, 5/2) and notice that 2x x3 > 0 for small x, e.g. x = .
PSTAT 120A, Spring 2010, Homework Assignment 4. Due Wednesday April 28 in lecture. No late homework. Reading: Ross Sections 4.1-4.6, 4.7 up to and including example 7c, 4.8.1, 4.8.2 and 4.8.3 Give a detailed solution with an argument, not just a numerical
Solution to HW4
April 29, 2010
4.5 by dene X = the dierence between # of heads and # of tails in ips. When n heads, 0 tails: X=n; when n-1 heads, 1 tails: X=n-2; ; when 0 heads, n tails: X=-n; so we conclude that X can take n possible values in cfw_n 2i,
PSTAT 120A, Spring 2010, Homework Assignment 3. Due Wednesday April 21 in lecture. No late homework. Reading: Ross Sections 3.2-3.4 (up to and including example 4i) Complete and clear working must be shown to receive credit. A numerical nal answer is requ
PSTAT 120A Spring 2010 HW 3 Solutions
1. (Ross 3.15) Note that percentages represented as a decimal between 0 and 1 will obey essentially the same properties as probabilities, because if a person is picked randomly, then the probability that they have som
PSTAT 120A, Spring 2010, Homework Assignment 2. Due Wednesday April 14 in lecture. No late homework. Reading: Ross Sections 2.2-2.4 (up to and including example 4a), 2.5 (up to and including example 5l) Complete and clear working must be shown to receive