36-225 Handout for Lecture # 9
September 15, 2008
1. A new method (method B) for teaching a certain industrial skill was developed, and it was found that there is 90% chance of successfully learning the skill when this method is used. Even though there is
36-225 Handout for Lecture # 7
September 10, 2008
1. The information you get with a certain prescription drug states: There is 10% chance of experiencing headache (H) There is 15% chance of experiencing nausea (N) There is 5% chance of experiencing both s
36-225 - Handout for Lecture # 5 September 5, 2008 Last time.
Axioms of Probability 1. P(A) 0 2. P(S) = 1 3. A B = = P(A B) = P(A) + P(B) If S has an infinite number of outcomes. 4. Let A1, A2, A3, . be events such that Ai Aj = for each i = j (pairwise di
Lecture #1 (8/25/08)
STAT 36-225, Fall 2008
I. Probability is not always intuitive.
1. Let's Make a Deal As a motivating example behind the discussion of probability, we'll investigate the Let's Make a Deal Paradox. This paradox is related to a popular te
36-225 Introduction to Probability Theory
Fall 2015
Instructor
Peter Freeman
Office
BH 229G
Email
[email protected]
9:30 AM
10:20 AM
Office Hours
Tu 3-5 and Th 2-5
(or by appointment)
Lecture
MWF
Doherty 2315
Required Text
Mathematical Statistics with Appl
1. (8 points) Let U = 21.11 Y, where the Y,- are iid draws from a Geometric(0.5) distribution. What
is the variance of U?
Viv] %1Vl
eta] Viki-.3
Viol
2. (8 points) You are to sample data randomly from a distribution P with mean ,u and standard deviation
0
1. (8 points) Let U = i 2; Yi, Where the Y; are iid draws from a Uniform(0,1) distribution. What is
the variance of U ?
\ILU] VY17\%;b;] VEY
Viall VEGl viml = iii
(a)
2. (8 points) You make 64 _iid draws from a distribution With mean 10 and standard devia
1. (8 points) Let U = zz; 1, where the Y,- are iid draws from a Gamma(2,1) distribution. What is
the variance of U?
2. (8 points) You make 81_iid draws from a distribution with mean 10 and standard deviation 9. What
is the probability that Y is less than
Tear @ SOLUUON SET
1. (6 points overall) Warmup question: assume you are working with a gamma distribution with
E[Y] = 5 and V[Y] = 10. (a) (4 points) What are the values of a and ? (b) (4 points) What other
name is given to this particular distribution,
TE 5r SOLQTDN SE T
non-1am
1. (4 points) Warm-up question. You are givenlprobabilities for events A and B. You can use these to
compute P(A ['1 B) if and only if you make what assumption about the two events?
P(r3r\'>) :1 PGAB'PCB) i-i an only l; G and B
TEST CED Soc-anon SET
1. (6 points overall) Warmup question: assume you are working with a gamma distribution with
E[Y] = 3 and V[Y] = 9. (a) (4 points) What are the values of a and 6? (b) (4 points) What other
name is given to this particular distributio
Fig 51' $0Luou~l SET
1. (6 points overall) Warm-up question: assume you are working with a gamma distribution with
E[Y] = 3 and V[Y] = 9. (a) (4 points) What are the values of a and ,6? (b) (4 points) What other
name is given to this particular distributi
TE$T (B) Soufe S ET"
1. (4 points) Warm-up question. You are given (nonzero!) probabilities for events A and B. You can use these
to compute P(A n B) if and only if you make What assumption about the two events?
PC9033 = P(A) 9(3) 3: one only cfw_5? H mn
TEST SOLunoM SET
1. (4 points) Warmup question. You are given (nonzero!) probabilities for events A and B. You can
use these to compute P(A n B) if and only if you make what assumption about the two events?
?(An 5) = P(A)PCB) W an oak/18" Pi and B areinde
15-150 Spring 2015
Homework 06
Out: Wednesday, 18 February 2015
Due: Tuesday, 24 February 2015 at 23:59 EST
NOTE: Since this hw is due right before the midterm, late days are not
permitted for this hw
1
Introduction
This homework will focus on higher orde
15-150 Spring 2015
Homework 01
Out: Wednesday, 14 January 2015
Due: Tuesday, 20 January 2015 at 23:59 EST
1
Introduction
Welcome to 15-150! This assignment introduces the course infrastructure and the SML
runtime system, then asks some simple questions re
36-225 Introduction to Probability Theory Fall 2015
Homework 8
Due Friday, November 6th at 9:30 AM in DH 2315
Exercise numbers refer to Wackerly 7. If you are not using the American print edition of Wackerly 7:
check the problem scans on Blackboard to ens
36-225 Handout for Lecture # 11
September 19, 2008
1. Consider the following two random variables: Y - the daily number of defective products coming out of production line A. X - the daily number of defective products coming out of production line B. Y an
36-225 - Handout for Lectures # 13 & 14 September 24 & 26, 2008 Special Discrete Random Variables (Distributions)
I. The Binomial Random Variable (Distribution) Consider an experiment which consists of n independent (Bernoulli) trials, each having two pos
36-225 - Handout for Lecture # 16 October 1, 2008
Special Discrete Random Variables - Review
I. The Binomial Random Variable X Bin(n, p). X counts the number of successes out of n independent (Bernoulli) trials, each having a probability of "success" p. I
36-225 Handout for Lecture # 28
October 29, 2010
1. In the game of roulette a wheel with 38 numbered spaces (18 red, 18 black, 2 green) is spun and ball drops into one of the slots. Suppose Player 1 bets $10 on "Red" and Player 2 bets $10 that the result
36-225 Handout for Lecture # 26
October 25, 2010
Beta Densities ( = ) 4 f(x) 0 0.0 1 2 3
0.2
0.4 x
0.6
0.8
1.0
Beta Densities ( < ) 4 f(x) 0 0.0 1 2 3
0.2
0.4 x
0.6
0.8
1.0
1. The saturation of dissolved oxygen in a particular river can be modeled using a
36-225 Handout for Lecture # 23
October 18, 2010
1. Suppose scores on the verbal or math portion of the SAT from a particular year are approximately normally distributed with mean = 500 and standard deviation = 100. What is the proportion of students who
STAT 225 - Homework 9
Due Friday, November 5
Comments: 1. This homework is due Friday, November 5 in class, BEFORE class starts. 2. Please remember staple if you turn in more than one page. 3. You must always show all work. If you do not show your work, y
STAT 225 - Homework 8
Due Friday October 29
Comments: 1. This homework is due Friday, October 29 in class, BEFORE class starts. 2. Please remember staple if you turn in more than one page. 3. You must always show all work. If you do not show your work, yo
STAT 225 - Homework 7
Due Friday October 22
Comments: 1. This homework is due Friday, October 22 in class, BEFORE class starts. 2. Please remember staple if you turn in more than one page. 3. You must always show all work. If you do not show your work, yo
STAT 225 - Homework 6
Due Monday October 18
1. This homework is due Monday, October 18 in class, BEFORE class starts. 2. Please remember staple if you turn in more than one page. 3. You must always show all work. If you do not show your work, you may not