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### Sum of two independent Normal random variables 11/25

Suppose we consider and , both indepedent, I showed that through rotational symmetry of the joint distribution of (U,V) any change of coordinate system, by a rotation of the first axis for instance would also give a Normal(0,1) random variable: thus for any numbers and such that , we take the angle such that and , and the linear combination of will be Normal(0,1).

As a consequence, the sum of two independent standard Normals will be Normal(0,2). (Taking More generally suppose we want to consider the sum of two independent Normals that are not standardized: We know through the chapter on expectations and variances that the sum of these two independent random variables will have expectation and variance . Now we consider the standardized random variable this can be rewritten as the sum : which is of the form , with , and U and V independent standard normals.    Next: Limit Theorems Up: Sums of Continuous Random Previous: Gamma density
Susan Holmes
1998-12-07