fnl_sol

fnl_sol - 1 EE464 - Chugg, Spring 1994 - Final Exam...

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Unformatted text preview: 1 EE464 - Chugg, Spring 1994 - Final Exam Solution 1 Basic Problems (60 points) (1a) (10 points) In general P ( A B ) = P ( A ) + P ( B ) P ( A B ), so that P ( A B ) = P ( A ) + P ( B ) if and only if P ( A B ) = 0. A sufficient condition is that A and B are mutually exclusive events (i.e. A B = ). The cumulative distribution function (cdf) of X ( u ) is defined as F X ( u ) ( z ) = Pr { X ( u ) z } . The probability density function (pdf) of X ( u ) is related to the cdf by f X ( u ) ( z ) = d dz F X ( u ) ( z ) (1b) (10 points) These are all basic definitions: m X = integraldisplay - xf X ( u ) ( x ) dx 2 X = integraldisplay - ( x m X ) 2 f X ( u ) ( x ) dx = integraldisplay - x 2 f X ( u ) ( x ) dx m 2 X Pr { a < X ( u ) b } = integraldisplay b a f X ( u ) ( x ) dx (1c) (10 points) X ( u ) is uniform between a = 1 and b = 4 m X = a + b 2 = 5 / 2 2 X = ( b a ) 2 12 = 3 / 4 Pr { X ( u ) (0 , 2] } = integraldisplay 2 f X ( u ) ( x ) dx = integraldisplay 2 1 dx 3 = 1 / 3 . (1d) (10 points) We have discussed many special properties of Gaussian random variables; here are a few: The complete statistical description (i.e., the pdf) depends only on the second moment description (i.e., means, variances and covariances). A linear combination of jointly-Gaussian random variables produces jointly-Gaussian random variables. 2 K.M. Chugg - May 3, 1994 Two jointly-Gaussian random variables are independent if and only if they are uncorrelated. If X ( u ) and Y ( u ) are jointly-Gaussian, then Y ( u ) is Gaussian conditioned on the value of X ( u ) and vice-versa. The conditional mean is an affine function: m Y | X ( x ) = m Y + Y X ( x m X ) . The conditional variance of Y ( u ) given X ( u ) = x is not a function of x . The distribution of a normalized sum of independent identically distributed ran- dom variables tends to Gaussian as the number of random variables in the sum tends to infinity. This is the Central Limit Theorem - or as we studied it, the approximation to the Binomial distribution. (1e) (10 points) FALSE: In general, if two random variables are uncorrelated, they are not inde- pendent. TRUE: If X ( u ) and Y ( u ) are independent, then W ( u ) = [ X ( u )] 3 and Z ( u ) = cos(2 Y ( u )) are also independent and thus uncorrelated. FALSE: The second moment description does not determine the complete statis- tical description. FALSE: The joint behavior of X ( u ) and Y ( u ) is not determined by the marginal densities, but rather by the joint density. FALSE: This is seen by Jensens inequality, a special case of which ensures that the variance is non-negative: 2 X = E { [ X ( u )] 2 } m 2 X ....
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fnl_sol - 1 EE464 - Chugg, Spring 1994 - Final Exam...

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