In this short article the following inequality called the “Pitman inequality” is proved for the exchangeable random vector (*X*_{1},*X*_{2},…,*X*_{n})(X1,X2,…,Xn) without the assumption of continuity and symmetry for each component *X*_{i}Xi:

P(|1n∑i=1nXi|≤|∑i=1nαiXi|)≥12 ,

where allαi≥0 are special weights with∑i=1nαi=1.

]]>These results are consequences of — and discovered as a result of — an analysis of varieties of "strict" restriction semigroups, namely those generated by Brandt semigroups and, more generally, of varieties of "completely r-semisimple" restriction semigroups: those semigroups in which no comparable projections are related under the generalized Green relation 𝔻. For example, explicit bases of identities are found for the varieties generated by B_{0} and B_{2}.