3 
Functions of Random Variables and Geometric Probability 

3.1 
Functions of Random Variables: Derived Distributions 

3.2 
Perturbation Methods 
 
3.2.1 
Perturbations to a Random Variable 
 
3.2.2 
Perturbations to a PDF 
 
3.2.3 
Perturbations to a Sample Space 

3.3 
Geometric Probability 
 
3.3.1 
Buffon's Needle Experiment 
 
3.3.2 
Bertrand's Paradox 
 
3.3.3 
Cauchy Distribution 

3.4 
Assumption of Uniformity 
 
3.4.1 
A Geometrical Probability Interpretation of PDF's 

3.5 
Crofton's Method for Computing Mean Values 
 
3.5.1 
Response Distance of an Ambulette Revisited 
 
3.5.2 
Summary of Method 

3.6 
Coverage 

3.7 
Expected Travel Distances and Times: Some Practical Results 
 
3.7.1 
Simple Model 
 
3.7.2 
More Realistic TravelTime Model 
 
3.7.3 
Expected Travel Distances: The General Case 

3.8 
Spatial Poisson Processes 
 
3.8.1 
Description and Postulates 
 
3.8.2 
Timespace Poisson Process 
 
3.8.3 
Application to Facility Location and Districting 

3.9 
Alternative Spatial Processes 
 
3.9.1 
Spread Process Yielding the Binomial PMF 
 
3.9.2 
Clustered Process Yielding the Negative Binomial PMF 

3.10 
Conclusion 
*** 
References 
*** 
Problems 