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A random variable is said to have elliptical distribution if its probability density is a function of a quadratic form. This class includes the Gaussian and many other useful densities in statistics. It is shown in this paper that this class of densities can be expressed as integrals of a set of Gaussian densities. This property is not changed under linear transformation of the random variables. It is also proved in this paper that the conditional expectation is linear with exactly the same form as the Gaussian case. Many estimation results of the Gaussian case can be readily extended. Problems of computing optimal estimation, filtering, stochastic control, and team decisions in various linear systems become tractable for this class of random processes.