Recitation7Sols - Rensselaer Electrical, Computer, and...

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Rensselaer Electrical, Computer, and Systems Engineering Department ECSE 4500 Probability for Engineering Applications Recitation 7 Solutions Wednesday March 31, 2010 CommonMeansandVar iances — continuous RVs RV type pdf f ( x ) mean μ variance σ 2 uniform U ( a, b ) 1 2 ( a + b ) 1 12 ( b a ) 2 exponential 1 μ e x/μ u ( x ) μμ 2 Gaussian 1 2 πσ e ( x μ ) 2 2 σ 2 μσ 2 Laplacian 1 2 σ e 2 σ | x | 0 σ 2 Rayleigh x σ 2 e x 2 2 σ 2 u ( x ) p π 2 σ ¡ 2 π 2 ¢ σ 2 Solutions: Uniform We assume that b>a here. For the mean, μ = Z b a 1 b a xdx = 1 b a μ 1 2 x 2 | b a = 1 2 1 b a ¡ b 2 a 2 ¢ = 1 2 ( b + a ) . For the variance, we f rst calculate E [ X 2 ] and then use σ 2 = E [ X 2 ] μ 2 . For X U ( a, b ) , E [ X 2 ]= Z b a 1 b a x 2 dx = 1 b a μ 1 3 x 3 | b a = 1 3 1 b a ¡ b 3 a 3 ¢ = 1 3 1 b a ( b a )( b 2 + ab + a 2 ) = 1 3 ( b 2 + ab + a 2 ) . 1
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Hence σ 2 = E [ X 2 ] μ 2 = 1 3 ( b 2 + ab + a 2 ) μ 1 2 ( b + a ) 2 = 1 12 ( b a ) 2 . Note: One can simplify the math here somewhat by shifting the interval ( a, b ) to start at the origin, and then adding a back in to get the correct mean μ. Thevar iancew i l lnotbe a f ected by the shift, so no correction is needed there.
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This note was uploaded on 04/15/2010 for the course ECSE 4500 taught by Professor Woods during the Spring '08 term at Rensselaer Polytechnic Institute.

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Recitation7Sols - Rensselaer Electrical, Computer, and...

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