Definition 7 the covariance of x and y are is defined

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Definition 7 The covariance of X and Y are is defined as cov( X, Y ) = E [( X - E [ X ])( Y - E [ Y ])] The variance of X is defined as var( X ) = cov( X, X ) . 2 4. cov( X, Y ) = E [ XY ] - E [ X ] E [ Y ] . var( X ) = E [ X 2 ] - ( E [ X ]) 2 . 5. If X and Y are independent then cov( X, Y ) = 0 . If cov( X, Y ) = 0 , we say that X and Y are uncorrelated. Again, uncorrelated does not imply independence. 6. var( X + Y ) = var( X ) + var( Y ) + 2 cov( X, Y ) . If cov( X, Y ) = 0 then var( X + Y ) = var( X ) + var( Y ) . 7. cov( aX + b, cY + d ) = ac cov( X, Y ) for all constants a, b, c, d R . Thus var( aX ) = a 2 var( X ) . Definition 8 If 0 < var( X ) < and 0 < var( Y ) < , the correlation coeffiicent between X and Y is ρ ( X, Y ) = cov( X, Y ) p var( X ) var( Y ) . This is a normalized version of the covariance. 2
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ECE 6010: Lecture 2 – More on Random Variables 8 8. | ρ | ≤ 1 . This can be shown using the Cauchy-Schwartz inequality. | ρ | = 1 iff X and Y are linearly related, X = aY + b for some constants ( a, b ) with a 6 = 0 . Example 3 If ( X, Y ) ∼ N ( μ x , μ y , σ 2 x , σ 2 y , ρ ) , then ρ ( X, Y ) = ρ . 2 As we have observed before, if X, Y are jointly Gaussian and ρ = 0 , then they are independent. Otherwise, ρ = 0 does not imply independence. Characteristic functions The characteristic function is essentially the Fourier transform of the p.d.f. or p.m.f. They are useful in practice not for the usual reasons engineers use Fourier transforms (e.g., fre- quency content), but because they can provide a means of computing moments (as we will see), and they are useful in finding distributions of sums of independent random variables. Definition 9 Let X be a r.v. The characteristic function (ch.f.) of X is φ X ( u ) = E [ e iux ] for u R . (Here, i = - 1 . We will not use - 1 = j .) 2 Let us write some more explicit formulas. Suppose X is a continuous random variable. Then (by the law of the unconcious statistician) φ X ( u ) = Z -∞ e iux f X ( x ) dx. This may be recognized as the Fourier transform of f X ( x ) , where u is the “frequency” variable. (Comment on sign of exponent.) Note that given φ X we can determine f X by an inverse Fourier transform: If X is a discrete r.v., φ X ( u ) = X i e iux i p X ( x i ) , which we recognize as the discrete-time Fourier transform, and as before u is the “fre- quency” variable. (Comment on the sign of the exponent.) Given a φ X , we can find p X by the inverse discrete-time Fourier transform. Properties: 1. φ X (0) = 1 . (Why?) 2. | φ X ( u ) | ≤ 1 u . (Why?) 3. φ X and f X form a unique Fourier transform pair. f X φ X . Thus, φ X provides yet another way of displaying the probability structure of X . 4. φ X ( u ) = R -∞ e iux dF X ( x ) . This is referred to as the Fourier-Stieltjes transform of F X . 5. φ X is uniformly continuous.
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ECE 6010: Lecture 2 – More on Random Variables 9 Definition 10 For an r.v. X , the k th moment of X is E [ X k ] , for k N . 2 We can write E [ X k ] = Z -∞ x k dF X ( x ) . Theorem 1 If E [ | X | k ] < then E [ X k ] = i - k d k du k φ X ( u ) u =0 . That is, we can obtain moments by differentiating the characteristic function. For this rea- son, characteristic functions (or functions which are very similarly defined) are sometimes referred to as moment generating functions .
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  • Fall '08
  • Stites,M
  • Probability theory, lim

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