**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 *m _{X}*=E

c.
Calculate the variance of *X *(i.e. calculate var* _{X}*=E

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, m _{X}=*E

f.
Calculate the mean of *Y, m _{Y}*= E

g.
Calculate the variance of *X, *var* _{X}=*E

h.
Calculate the variance of *Y, *var_{Y}

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

j.
Calculate the correlation between* X* and *Y, **r _{XY}=*

k. Calculate the marginal pdf of *f _{XY} (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 *c** _{X}(r)* for a homogeneous spatial random field

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.).