For uniformly elliptic operators in $\mathbf R^n$ whose coefficients are smooth random functions which are homogeneous with respect to shifts in $\mathbf R^n$ the concept of an index is introduced and its properties are investigated. The index of a family of such operators is also defined. Formulas are established which express the index of an operator or a family of operators in terms of the principal symbol. Examples and special cases are considered. Bibliography: 9 titles.