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handout_week9_a - Stochastic Signals and Systems Multiple...

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Stochastic Signals and Systems Multiple Random Variables Virginia Tech Fall 2008 Jointly Gaussian RVs Let X and Y be independent RVs with the distributions: X is N ( 0 2 ) . Y takes the value 1 with probability p and - 1 with probability q = 1 - p . Deﬁne W = XY . (a) Show that W is Gaussian. (b) Find the joint distribution of X and W . Are these variables jointly Gaussian?

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Central Limit Theorem Let S n be the sum of n independent, identically distributed random variables with ﬁnite mean E [ X ] = μ and ﬁnite variance σ 2 , and let Z n be the zero-mean, unit-variance random variable deﬁned by Z n = S n - n μ σ n then lim n →∞ P [ Z n z ] = 1 2 π Z z -∞ e - x 2 / 2 dx . In other words, F Z n ( z ) converges to the standard normal cdf. The ubiquity of Gaussian rvs in science and engineering can be attributed in part to the fact that errors in measurements and noise in systems are often the sums of great many small, random integrants, and by virtue of the central limit theorem, the distributions of such
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handout_week9_a - Stochastic Signals and Systems Multiple...

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