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 variables. The distribution of
2 =
n
n
2
Zi
i=1
is call
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 of course to maximize gains with a
minimum of uncertain
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 community doubles in size during a unit time period,
for ex
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
otherwise
Find the CDF of Y = |X|.
Problem 2 (33 points)
A
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 appropriate units)
You intend to use beam 1 for construction.
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 in the figure below. (Notice that the
distribution is
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 two random loads W1 and W2, i.e.
WW
S = 1 + 2 where A i
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 and variance of Y.
c) Approximate the mean value and va
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 materials are transported by highway, 35% by
rail, and 15% by
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 minute is 10-5
and the probability of a moderate earthquake
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. Because of a gas shortage, there is a
probability of 0.3 th
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 the following:
For a given suburb of Boston and a certai
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 exponential PDF
fX(x) = e-x, x 0
Find and plot the PDF of the
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 ) = $ 8
%0
elsewhere
&
a) Find and plot the marginal probabil
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 you find P[A B] ?
Given that A and B are independent:
a)
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 shown below:
1
2
1
3
3
4
2
4
Configuration 1
Configuration
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 limit theorem establishes that, under mild conditions on F
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 m2, variances
12 and 22, and correlation coefficient , t
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-Moment Characterization
Mean (expected value) of a ran
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), nd the
(derived) distribution of the random variable Y
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 of an
X 2
experiment to the (x1, x2) plane.
Discrete
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 probability density in an interval [x1 , x2 ]
P [x1 < X x2
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.
PX (x) = 1
all x
Cumulative Distribution Function (C
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)
Sample Space, S: collection of all possible outcomes (s
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 appropriate units)
You intend to use beam 1 for construction.
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 in the figure below. (Notice that the
distribution is
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
otherwise
Find the CDF of Y = |X|.
Problem 2 (33 points)
A
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 community doubles in size during a unit time period,
for ex
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 you find P[A B] ?
Given that A and B are independent:
a)
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 shown below:
1
2
1
3
3
4
2
4
Configuration 1
Configuration