1.010 - Brief Notes # 9
Point and Interval Estimation of Distribution Parameters
(a) Some Common Distributions in Statistics
.
Chi-square distribution
Let Z1 , Z2 , . . . , Zn be iid standard normal
1.010 Fall 2008
Homework Set #9
Due November 20, 2008 (in class)
1. Let X1 and X2 be the gains, in millions of dollars, from investing $1million in two
different stocks. The objective of investing is
1.010 Mini-Quiz #2
(45 min open books and notes)
Problem 1 (40 Points)
A community of bacteria initially includes 1000 individuals. Given favorable conditions
(light, temperature, nutrients), the comm
1.010 Mini-Quiz #3
(40 min open books and notes)
Problem 1 (33 points)
A random variable X has uniform distribution between 1 and 1. This means that the
PDF of X is
0.5,
fX (x) =
0,
for 1 < x < 1
oth
1.010 Mini-Quiz #5
(45 min open books and notes)
Problem 1 (60 Points)
Two concrete beams have strengths X1 and X 2 with joint normal distribution:
3 1 0.75
X1
X ~ N 3, 0.75 1
2
(in some approp
1.010 Mini-Quiz #4
(40 min open books and notes)
Problem 1 (30 Points)
Consider two variables X1 and X 2 , each with possible values 1 and 1. The joint
probability mass function of X1 and X 2 is shown
1.010 Fall 2008
Homework Set #8
Due November 13, 2008 (in class)
!
1. A concrete column has square cross-section with side length 0.5m. The column is
subject to a random axial stress S contributed by
1.010 Fall 2008
Homework Set #7
Due November 6, 2008 (in class)
1. X has uniform distribution between 0 and 1. Let Y = X 2.
a) Find the exact mean value and variance of X
b) Find the exact mean value
1.010 Fall 2008
Homework Set #2
Due September 25, 2008 (in class)
1. There are three modes of transporting materials from Boston to Chicago: by plane, by
highway, and by rail. About half of the materi
1.010 Fall 2008
Homework Set #1
1. If the occurrences of earthquakes and high winds are unrelated, and if, at a particular
location, the probability of a high wind occurring throughout any single minu
1.010 Fall 2008
Homework Set #3
Due October 2, 2008 (in class)
1. The service stations along a highway are located according to a Poisson process with
an average of 1 service station in 10 miles. Beca
1.010 Fall 2008
Homework Set #4
Due October 9, 2008 (in class)
1. Calculate and plot the hazard function for the lifetime distribution shown below.
fT(t)
1
2
0
t
2. Read Application Example 7 and do t
1.010 Fall 2008
Homework Set #6
Due October 28, 2008 (in class)
1. A chain has 10 links. The strengths of the links, X1, , X10 are independent and
identically distributed random variables with exponen
1.010 Fall 2008
Homework Set #5
Due October 16, 2008 (in class)
1. Two continuous variables X and Y have joint probability density function:
#1
% ( x + y ) 0 " x " 2 and 0 " y " 2
f X ,Y ( x, y ) = $
1.010 Mini-Quiz #1
(30 min open books and notes)
Problem 1 (30 points)
Events A and B have probability P[A] = P[B] = 0.25.
Given that A and B are mutually exclusive:
a) Can you find P[A B] ?
b) Can yo
1.010 Final Exam
(3 hours open books and notes)
Problem 1 (10 Points)
Christmas lights sold at Wall-Mart come in sets of 4 bulbs of different colors connected
in two alternative configurations as show
1.010 - Brief Notes # 8
Selected Distribution Models
The Normal (Gaussian) Distribution:
Let X1 , . . . , Xn be independent random variables with common distribution FX (x). The so called
central lim
1.010 - Brief Notes #7
Conditional Second-Moment Analysis
Important result for jointly normally distributed variables X1 and X2
If X1 and X2 are jointly normally distributed with mean values m1 and m
1.010 - Brief Notes # 6
Second-Moment Characterization of Random Variables and Vectors.
Second-Moment(SM) and First-Order Second-Moment(FOSM)
Propagation of Uncertainty
(a) Random Variables
.
Second-
1.010 - Brief Notes # 5
Functions of Random Variables and Vectors
(a) Functions of One Random Variable
Problem
Given the CDF of the random variable X, FX (x), and a deterministic function Y = g (x),
1.010 - Brief Notes #4
Random Vectors
A set of 2 or more random variables constitutes a random vector. For example, a random
X
vector with two components, X = 1 , is a function from the sample space
1.010 - Brief Notes # 3
Random Variables: Continuous Distributions
Continuous Distributions
Cumulative distribution function (CDF)
FX (x) = P [X x]
P [x1 < X x2 ] = FX (x2 ) FX (x1 )
Average prob
1.010 - Brief Notes # 2
Random Variables: Discrete Distributions
Discrete Distributions
Probability Mass Function (PMF)
PX (x) = P (X = x) =
P (O)
all O: X (O )=x
Properties of PMFs
1. 0 PX (x) 1
2
Brief Notes #1
Events and Their Probability
Definitions
Experiment: a set of conditions under which some variable is observed
Outcome of an experiment: the result of the observation (a sample point)
1.010 Mini-Quiz #5
(45 min open books and notes)
Problem 1 (60 Points)
Two concrete beams have strengths X1 and X 2 with joint normal distribution:
3 1 0.75
X1
X ~ N 3, 0.75 1
2
(in some approp
1.010 Mini-Quiz #4
(40 min open books and notes)
Problem 1 (30 Points)
Consider two variables X1 and X 2 , each with possible values 1 and 1. The joint
probability mass function of X1 and X 2 is shown
1.010 Mini-Quiz #3
(40 min open books and notes)
Problem 1 (33 points)
A random variable X has uniform distribution between 1 and 1. This means that the
PDF of X is
0.5,
fX (x) =
0,
for 1 < x < 1
oth
1.010 Mini-Quiz #2
(45 min open books and notes)
Problem 1 (40 Points)
A community of bacteria initially includes 1000 individuals. Given favorable conditions
(light, temperature, nutrients), the comm
1.010 Mini-Quiz #1
(30 min open books and notes)
Problem 1 (30 points)
Events A and B have probability P[A] = P[B] = 0.25.
Given that A and B are mutually exclusive:
a) Can you find P[A B] ?
b) Can yo
1.010 Final Exam
(3 hours open books and notes)
Problem 1 (10 Points)
Christmas lights sold at Wall-Mart come in sets of 4 bulbs of different colors connected
in two alternative configurations as show