CEE 304 – Section 4 Problems (9/20/2006) 1. Transforming Variables Example: ( )xXfxe−=for x > 0, and 0 otherwise Develop the density function for Y = X ½Y = g(X) = X ½=> X = Y2, ydydx2=One-to-one transformation: we have a one-to-one relationship between X and Y, so we can use the following formula to find the pdf of Y given the pdf of X: ( )( )( )( )dydxyxfdydxdxyxdFdyydFyfXXYY⋅===2( )(2 )yYfyey−∴=⋅for y > 0, and 0 otherwise 2. Memoryless Exponential Process: ( )xXfxeλλ−=for x≥0( )1xXFxeλ−=−for x> 0 By definition: ( )XFxP Xx=≤1( )xXP XxFxeλ−∴≥=−=Additional lifetime (i.e. how much long it will last) = A ttaXXeetFtaFtXPtXtaXPtXtaXPλλ−+−=−+−=≥≥∩+≥=≥+≥)()(1)(1]|[aeλ−=By definition: ()( )tXetFtXPλ−=−=≥1, for t ≥0Therefore, the distribution of the additional lifetime A is exactly the same as the original distribution of waiting time T. So, E[A] = 1/λ. 1
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