Vs x and y let v minx y 1 fv v p v v p

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Unformatted text preview: FZ (z ) = P (max(X , Y ) ≤ z ) = P (X ≤ z )P (Y ≤ z ) PDF of z is given by fZ (z ) = dFZ dz (z ) Consider two independent r.v.s X and Y , let V = min(X , Y ) 1 − FV (v ) = P (V ≥ v ) = P (min(X , Y ) ≥ v ) = P (X ≥ v )P (Y ≥ v ) PDF of v is given by fZ (z ) = M. Chen ([email protected]) dFZ dz (z ) ENGG2430C lecture 9 11 / 20 Derived Distribution: Example * Consider two independent r.v.s X ∼ Exp (µx ) and Y ∼ Exp (µy ), find the PDF of V = min(X , Y ): for all v ≥ 0, we have P (V ≥ v ) = P (X ≥ v )P (Y ≥ v ) ˆ∞ ˆ∞ = µx e −µx x dx µy e −µy y dy v v = e −µx v e −µy v = e −(µx +µy )v As such, FV (v ) = 1 − e −(µx +µy )v , and fV (v ) = M. Chen ([email protected]) (µx + µy )e −(µx +µy )v , if v ≥ 0; ; 0, otherwise. ENGG2430C lecture 9 12 / 20 Derived Distribution: Strictly Monotonic Cases Let X be an r.v. and Y = g (X ); g (·) is a strictly monotonic function. (From MIT opencourse 6.041 slides.) Let y = g (x ), we have {x ≤ X ≤ x + δ } = {g (x ) ≤ Y ≤ g (x ) + δ | f X (x )δ = fY (g (x ))δ | dg (x )|} dx dg (x )| dx In conclusion, fY (y ) = fX g −1 (y ) /| dg (g −1 (y ))| dx M. Chen ([email protected]) ENGG2430C lecture 9 13 / 20 Example: Y = aX + b (From MIT opencourse 6.041 slides.) Assuming...
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This document was uploaded on 03/31/2014.

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