Calculus useful for Statistics
1. The set of real numbers
x R is a real number.
Examples:
x=2,
x=1/3=0.333333
2. Function
f(x) is a function which, for any x R, associates a unique value f(x).
Example:
3. Derivatives
df ( x)
dx
x = x0
=f(x0) is the deriva

Probability and Statistics > Statistical Distributions > Continuous Distributions
Normal Ratio Distribution
of independent normally distributed variates with zero mean is distributed with a Cauchy The ratio distribution. This can be seen as follows. Let a

Probability and Statistics > Statistical Distributions > Continuous Distributions
Normal Product Distribution
The distribution of a product of two normally distributed variates is given by
and
with zero means and variances
and
(1)
(2)
where is a delta fun

Space/Time mapping with Soft data
Soft Data
X(p)=X(s,t) is a S/TRF. Soft data for X(p) is available at the soft data points psoft = (p1,., pns). The vector of random variables xsoft = (x1,.,xns) represents the S/TRF at the soft data point, i.e. xsoft=(X(p

Combining Random Variables
1. Sum of two random variables
Let X and Y be two random variables (r.v.), let Z = X + Y be the r.v. defined as their sum, and let x, y and z be realizations of X, Y and
Z, respectively.
1.1 The pdf of the sum of two random vari

Propagation of uncertainty
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In statistics, propagation of uncertainty (or propagation of error) is the effect of
variables' uncertainties (or errors) on the uncertainty of a function based o

Adjusting for sampling variability in disease mapping
using a Bayesian soft data model
Background
The goal of this paper is the space/time mapping analysis of the HIV incidence rate
among the population tested across North Carolina, where the HIV incidenc

Adjusting for sampling variability in disease mapping
using a Poisson soft data model
1. Simple co-kriging
Suppose that the random variable xk and the random vector zh=[ zh1, , zhn]T are correlated with
known covariance / cross covariance. We seek the bes

Space/Time mapping
3. Space/time mapping using second order moments and hard data: The
Simple Kriging space case
We will consider a special case of the general BME framework corresponding to the case where the
general knowledge consists only of the mean t

Risk Assessment in Practice
EPIDEMIOLOGICAL STUDIES
Case-control studies begin by identifying patients with the outcome (disease) of interest
and looks backward (retrospective) to see if they had the exposure of interest. Cases,
people who have the outcom

Your name:_
ENVR130 Fall 2005 In-class Examination 2. Oct 31, 2005
Answer all questions on these sheets of paper. Use the reverse side if necessary. Show all
calculations and the reasoning supporting your answer.
Question 1
i. Which of the four reactions

Lecture 2 Notes
Lead is a heavy metal found in the air
Historically lead came motor vehicles. After phase out of leaded gasoline, a
major source is now lead smelters
People are exposed to lead by breathing, and it accumulates in the blood
Health impac

Lecture 3 Notes
A homogeneous/stationary S/TRF is defined by
A mean trend that is constant over space (homogeneity) and time
(stationarity)
mX(p) = mX
A covariance between point p =(s,t) and p =(s,t) that is only a function
of spatial lag r=|s-s| and th

Lecture 4 Notes
Bayesian soft data for the true incidence using observed incidence as prior
The soft datum probability density function (PDF) describes the uncertainty associated
with the true incidence rate xij given site specific knowledge S that includ

HW 1
Part 1:
Start MATLAB (preferably a recent version).
1) Use the help on the command line and explore the general functions of MATLAB
(using > help; then > help general). Find the command that List current variables.
Using the help for that command, de

HW 2 Finished
HwklPart4solution
- solution of homework 1 part 4
% Generate, if needed, the data files for the spatiotemporal field
% -echo off;
if exist('./datastg.dat')~=2,
rand('state',3);
randn('state',2);
nSim=1;
cMS=20*rand(30,2);
nMS=size(cMS,1);
tM

Exam 1 Review Guide
Natural processes contribute to the production and distribution of environmental pollutants in space-time.
Health processes denote the variables describing changes in the health state of a receptor (human body, organ, skin, etc.) or a

Exam 2 Review Guide
Definitions
The Space/Time Random Field (S/TRF) X(p) =X(s,t) is a random variable that is a
function of location s=(s1,s2) and time t.
The space/time covariance between space/time points p and p is
cX(p,p) = E[(X(p)- mX(p)(X(p)- mX(p)]

Final Exam Review Guide
Suppose that the random variable xk and the random vector zh=[ zh1, , zhn]T are
correlated with known covariance / cross covariance. We seek the best linear unbiased
estimatek for xk given measured values for zh.
In simple co-krigi

Intro Notes for ENVR 765
The course starts with an introduction to space/time Geostatistics of environmental monitoring data to create maps representing the
distribution of exposure to humans across space and time, and then investigates how the resulting

Quiz 1 Review Guide
The knowledge bases
The total knowledge base K about the mapping situation is divided between general
knowledge base G, and site-specific knowledge base S, so that K=GU S.
The general knowledge base G includes all knowledge bases that

The Gaussian Distribution
1. Univariate Gaussian distribution
A Gaussian random variable x is completely defined by its mean m and variance 2 , and its pdf is
,
The moments of the Gaussian pdf are given by
(See proof below)
Additionally the expected value