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 - 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
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 # 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 # 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 # 22
October 22, 2008
1. In a certain city the daily consumption of electric power (in millions of kilowatts hours) can be treated as having a gamma distribution with = 4 and = 2. A new power plant is being built. What should be
36-225 Handout for Lecture # 28
November 5, 2008
1. Suppose that the joint pdf of (X, Y ) is given by:
1 (xy + 1) 80
0<x<4 0<y<4 otherwise
fXY (x, y) =
0
(a) Show that fXY (x, y) is a legitimate density function. (b) Find P (X 1, Y 1) 2. A nut company m
36-225 Handout for Lecture # 31
November 12, 2008
1. Let (X, Y ) be jointly continuous with the following joint density: fXY (x, y) = x + y 0 x, y 1 0 otherwise
In this case, X and Y denote the proportions of time that two employees actually spent working
36-225 Handout for Lecture # 36
November 14, 2008
1. Let Y1 and Y2 be independent Poisson random variables with means 1 and 2 respectively. Find the (a) probability function of Y1 + Y2
(b) the conditional probability function of Y1 given that Y1 + Y2 = m
8/27/2008 Should be in Chapters 1 and 2 in the book + and before or after x doesn't matter Returns old value before it prints the new one for System.out.println(+x + " " + x); =>It gives you the old value if you do it after, and the new value if you do it
P ( A U B) = P (A) + P (B) P (A or B) P(A U B U C) = P(A) + P (B) + P (C) P(A or B) P (A or C) P (B or C) + P (A or B or C) Problem #4 from Sept. 12th handout: P (C1 OR (C2 and C3) = P(C1 OR C2) AND (C1 OR C3) = P (C1 OR C2) + P (C1 OR C3) P (C1 AND C2 AN
36-225 INTRODUCTION TO PROBABILITY & STATISTICS I Midterm II Practice Exam NOTE: The actual exam will also have some multiple choice questions You must
show your work
in order to get full or be considered for partial credit.
You may use a single sheet of
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
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 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 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
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
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 # 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 # 30
November 3, 2010
1. Consider the following joint p.d.f: PXY (x, y) = e-2 x+y x! y! x = 0, 1, 2, 3 . . . y = 0, 1, 2, 3 . . . > 0
(a) Find and recognize the marginal distributions PX (x) and PY (y). (b) Are X and Y independe