Following a review of the basic ideas in structural reliability, including signature-based representation and preservation theorems for systems whose components have independent and identically distributed (i.i.d.) lifetimes, extensions that apply to the comparison of coherent systems of different sizes, and stochastic mixtures of them, are obtained. It is then shown that these results may be extended to vectors of exchangeable random lifetimes. In particular, for arbitrary systems of sizes m < n with exchangeable component lifetimes, it is shown that the distribution of an m-component system's lifetime can be written as a mixture of the distributions of k-out-of-n systems. When the system has n components, the vector of coefficients in this mixture representation is precisely the signature of the system defined in Samaniego, IEEE Trans Reliabil R-34 (1985) 69-72. These mixture representations are then used to obtain new stochastic ordering properties for coherent or mixed systems of different sizes.

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