University of California, Los Angeles Department of Statistics Statistics 100A Exam 1 21 October 2010 Name: Instructor: Nicolas Christou
Problem 1 (15 points) Three cards are identical in form except that both sides of the first one are colored green, bot

University of California, Los Angeles Department of Statistics Statistics 100A Exam 1 23 October 2007 Name: Instructor: Nicolas Christou
Problem 1 (25 points) Answer the following questions: a. What is the probability that you and the person sitting next

University of California, Los Angeles Department of Statistics Statistics 100A Exam 2 13 May 2011 Name: Instructor: Nicolas Christou
Problem 1 (25 points) Answer the following questions: a. Let X (, ). Show that Y = cX follows (, c).
b. Let X N (, ). Find

University of California, Los Angeles Department of Statistics Statistics 100A Homework 3
EXERCISE 1 Use the binomial theorem (go back to your classnotes from the beginning of the course) to show that if n X b(n, p) then x=0 p(x) = 1. EXERCISE 2 New York

University of California, Los Angeles Department of Statistics Statistics 100A Combinatorial analysis Basic principle of counting: Suppose two experiments are to be performed. Then, if the first experiment can result in m outcomes and if for each outcome

University of California, Los Angeles Department of Statistics Statistics 100A Occupancy problem
Consider n = 7 cells and r = 7 marbles. In how many ways can the 7 marbles be distributed in the 7 cells? Please complete the table below. Occupancy numbers 1

Probability Rules
Elisa Long
UCLA Anderson School of Management
A. Probability Experiments
An experiment (eg, die roll, coin toss) is an activity or procedure that produces distinct,
well-dened possibilities called outcomes. Outcomes are the nest grain; t

Conditional Probability & BayesRule
Elisa Long
UCLA Anderson School of Management
In this class, we assume a Bayesian perspective when thinking about probabilities. In a
nutshell, this means that the probability of an uncertain event occurring is a degree

University of California, Los Angeles Department of Statistics Statistics 100A Exam 2 16 November 2010 Name: Instructor: Nicolas Christou
Problem 1 (25 points) Answer the following questions: a. Many people cancel their reservations at hotels at the last

University of California, Los Angeles Department of Statistics Statistics 100A Exam 1 23 October 2007 Name: Instructor: Nicolas Christou
Problem 1 (25 points) Answer the following questions: a. What is the probability that you and the person sitting next

1. Let V be a vector space with inner product (, ). (a) Show that if u, v V satisfy u + v = u v then u and v are orthogonal. (b) Suppose that v, w V instead satisfy v = 1, w = 2, and v + w = 1. Find (v, w). 2. Let V be a vector space and cfw_v1 , ., vk b

STAT 100A Review for midterm
Note: The following are the materials to be covered in the midterm.
1
Basic concepts
When an experiment is performed, the outcome can be random. The sample space is the set of all the possible outcomes. It is often denoted by

STAT 100A HWV Due next Wed in class
Problem 1: For a discrete random variable X , prove (1) E[aX + b] = aE[X ] + b. (2) Var[aX + b] = a2 Var[X ]. Problem 2: For two discrete random variables X and Y , if X and Y are independent, prove (1) E[X + Y ] = E[X

STAT 100A HWIV Due next Wed in class
Problem 1: For Z Bernoulli(p), calculate E[Z ]. Problem 2: For X Binomial(n, p), calculate E[X ]. Problem 3: For X Geometric(p), calculate E[X ]. Problem 4: Suppose we have a ve-letter alphabet, A, B, C, D, E, and thei

STAT 100 Homework II Answer Key
Problem 1: Let X and Y be the two numbers respectively. p(X > 3) = 1/3. p(X > 3|X + Y > 9) = 1. p(X = k|X Y | > 2) = p(X = k, |X Y | > 2)/p(|X Y | > 2). p(|X Y | > 2) = 12/36 = 1/3. p(X = 1, |X Y | > 2) = 3/36 = 1/12. So p(

STAT 100A HWI Solution
Problem 1: Suppose we ip a fair coin 4 times independently. (1) What is the sample space? A: The sample space consists of all the 24 = 16 sequences of heads and tails. (2) What is the set that corresponds to the event that the numbe

STAT 100A HWII Solution
Problem 1: If we ip a fair coin n times independently, what is the probability that we observe k heads? k = 0, 1, ., n. Please explain your answer. A: The probability is n /2n . The reason is that all the 2n sequences are equally l

STAT 100A HWIII Solution
Problem 1: Prove the following two identities (C stands for cause, E stands for eect), where cfw_Ci , i = 1, ., n partition the whole sample space. (1) Rule of total probability: P (E ) = n P (Ci )P (E |Ci ). i=1 A: P (E ) = P (n

STAT 100A HWIV Solution
Problem 1: Suppose we roll a biased die, the probability mass function is p(1) = .1, p(2) = .1, p(3) = .1, p(4) = .2, p(5) = .2, and p(6) = .3. Let X be the random number we get by rolling this die. (1) Calculate Var(X ). A: Var(X

STAT 100A HWV Solution
Problem 1: For both discrete and continuous cases, prove (1) E[a + bX ] = a + bE[X ]. A: For discrete case, let p(x) be the probability mass function of X . Then E[a + bX ] = (a + bx)p(x) = a x p(x) + b x xp(x) = a + bE[X ]. x For c

STAT 100A HWVI Solution
Problem 1: Suppose we ip a fair coin n times independently. Let X be the number of heads. Let k = n/2 + z n/2, or z = (k n/2)/( n/2). Let g (z ) = P (X = k ). 2 (1) Using the Stirling formula n! 2nnn en , show that g (0) 1 n . a b

STAT 100A HWIII Due next Wed in class
Problem 1: Suppose an urn has r red balls and b blue balls. We random pick a ball, and then we put three balls of the same color back to the urn. After that we randomly pick a ball again. (1) What is the probability t