## Statistics Papers

#### Document Type

Journal Article

#### Date of this Version

1997

#### Publication Source

Theory of Probability & Its Applications

#### Volume

41

#### Issue

2

#### Start Page

199

#### Last Page

209

#### DOI

10.1137/S0040585X97975435

#### Abstract

Given probability distributions F_{1} , F_{2} , . . ., F_{k} on **R** and distinct directions θ_{1}, . . ., θ_{k}, one may ask whether there is a probability measure μ on **R**^{2} such that the marginal of μ in direction θ_{j} is F_{j}, j = 1, . . ., k. For example for k = 3 we ask what the marginal of μ at 45° can be if the *x* and *y* marginals are each say standard normal? In probabilistic language, if *X* and *Y* are each standard normal with an *arbitrary* joint distribution, what can the distribution of X + Y or X - Y be? This type of question is familiar to probabilists and is also familiar (except perhaps in that μ is positive) to tomographers, but is difficult to answer in special cases. The set of distributions for Z = X - Y is a convex and compact set, *C*, which contains the single point mass Z ≡ 0 since X ≡ Y, standard normal, is possible. We show that *Z* can be 3-valued, Z=0, ±a for any *a*, each with positive probability, but *Z* *cannot* have any (genuine) two-point distribution. Using numerical linear programming we present convincing evidence that *Z* can be uniform on the interval [-ε, ε] for ε small and give estimates for the largest such ε. The set of all extreme points of *C* seems impossible to determine explicitly.

We also consider the more basic question of finding the extreme measures on the unit square with uniform marginals on both coordinates, and show that not every such measure has a support which has only one point on each horizontal or vertical line, which seems surprising.

#### Copyright/Permission Statement

Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

#### Keywords

marginal distributions, extreme point, Radon

#### Recommended Citation

Applegate, D.,
Reeds, J.,
Scheinberg, S.,
Shepp, L. A.,
&
Shor, P.
(1997).
Some Problems in Probabilistic Tomography.
*Theory of Probability & Its Applications,*
*41*
(2),
199-209.
http://dx.doi.org/10.1137/S0040585X97975435

**Date Posted:** 27 November 2017

This document has been peer reviewed.