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 - Brief Notes
3
Random Variables: Continuous Distributions
ontinuous Distributions
Cumulative distribution function (CDF)
FX (x) = P [X x]
P [x < X x2 ] = FX (x2 )
FX (x )
Average probability density in an interval [x , x2 ]
P [x < X x2 ]
FX (x2 )
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 .
G
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 FX ,
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Civil and Environmental Engineering
1.017 Computing and Data Analysis for Environmental Applications
Quiz2
Thursday,November7,2002
Pleaseanswerallquestionsonaseparatepiece(s)ofpaperwithyournameclearlyide
1.010 - Brief Notes
2
Random Variables: Discrete Distributions
iscrete
istributions
Probability Mass Function (PMF)
P
PX (x) = P (X = x) =
P (O)
all O: X(O)=x
Properties of PMFs
1. 0 PX (x) 1
P
2.
PX (x) = 1
all x
Cumulative Distribution Function (CDF)
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 rando
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
P
i=1
2
Zi
is call
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
X2
experiment to the (x1, x2) plane.
Discrete Rand