HW4 - 1 Proof ESC Given Z ~ N(0 1 we have the probability...

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1. Proof ESC Given Z ~ N(0, 1), we have the probability density function: = - f'Z 12πe 12Z2 Let ~ ( , ) x N DL σL2 , we have: = + x σLZ DL (1) = dx σLdZ (2) ( )= ( ) f x 1σLf' Z (3) The ESC is given by: = = ∞ - ESC x ROP x ROPfxdx with = + ROP DL ss Replace the formulation of ESC by (1), (2) and (3): = = + - - ( ) = = - ( ) =- ESC x ROP σLZ DL DL ss1σLf' Z σLdZ Z ssσL σLZ ssf' Z dZ = ∞ ( ) + = ( ) ssZ ssσL f' Z dZ σLZ ssσL Zf' Z dZ We have: = ∞ ( ) = > = -
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