Homework 4 BMElib track

Random variables and spatial covariance models

Date given: 10/06

Due: 10/15 noon

Problem 1

Consider the following function

a.       Determine the constant  to ensure that the function  is the pdf of a random variable x (i.e. apply the normalization constraint).

b.      Calculate the expected value of X (i.e. calculate mX=EX[X])

c.       Calculate the variance of X (i.e. calculate varX=EX [(X-m)2])

d.      Calculate the probability that 0.1<X<0.3

Problem 2

Two random variables X and Y have the following bivariate pdf:

e.       Calculate the mean of X, mX=EXY[X]=

f.       Calculate the mean of Y, mY= EXY[Y]

g.      Calculate the variance of X, varX=EXY[(X- mX)2].

h.      Calculate the variance of Y, varY

i.        Calculate the covariance between X and Y, cov(X,Y)

j.        Calculate the correlation between X and Y,  rXY= cov(X,Y)

k.       Calculate the marginal pdf of fXY (x,y) with respect to y

l.         Calculate conditional pdf of X given Y=y

m.     Calculate probability that X<0.5 given that Y=2

Problem 3

List two models for the covariance cX(r) for a homogeneous spatial random field X(s).  Provide the equation for the covariance models and explain what each parameter is.

Review the sample program covEstimationS.m.  Using the T/GIS dataset for your class project, model the spatial covariance for a given time aggregated period (for example a given year), making sure that you combine (e.g. average) the duplicate data for that period.  Provide a figure with a marker plot or color plot of the data, and a figure showing both the experimental covariance values and the spatial covariance model you have fit to those values.

Select a different (non overlapping) time aggregated period (for example a different year), model the spatial covariance of the data for that period, and provide the figures described above (i.e. a map of the data and a figure of the experimental covariance values together with the covariance model).  Provide a write up describing your analysis and the results you obtained (i.e. how you chose the classes of spatial lags for each time period, whether the two models are similar, what sort of variability is your covariance model physically describing, etc.).