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Functions of Two

# Functions of Two - FUNCTIONS OF TWO RANDOM VARIABLES...

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1 FUNCTIONS OF TWO RANDOM VARIABLES Maximum ( ) Define max , W XY = [ ] [ ] , Then ( ) P P ( , ) W F w W w X w and Y w F w w = ≤= When and are independent, ( ) ( ) ( ) . In this case, we have ( ) ( ) ( ) ( ) ( ) W W YX X Y F w F wF w f w f wF w = = + W=X W=Y

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2 Minimum ( ) Define min , W XY = [ ] [ ] , () P P () (,) W X Y F w W w X w or Y w F w F w F w w = ≤= + . When and are independent, ( ) ( ) ( ) ( ) ( ). In this case, we have ( ) ( ){1 ( )} ( )}. W X Y WX Y Y X X Y F w F w Fw F w f w f wF w f w = +− = −+ . W=X W=Y
3 Difference Let X and Y be independent exponential random variables with arrival rates 12 and λλ respectively. Define Z XY = . Find ( ). Z fz

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4 A hard way to solve this problem: { } { } 12 21 , 00 11 22 1 2 ( ) Pr Pr for 0 (,) 1 () Z XY shaded area xz yz xy zz Z Fz z X zY X z z f x y dxdy dx dy e e dy dx e e ee fz e λλ λ ∞+ −− = = −≤ + > = = + = +− ++ = = + ∫∫ 2 1 e + + yxz = + = z yx = z X Y 1 2
5 Physical Meaning 12 21 () 0 zz Z fz e e z λλ −− = +> ++ Homework: [ ] [ ] Show P = and P Y X XY ≤= When , z Z e λ = = = . Note the memoryless property of the Poisson arrival process.

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Functions of Two - FUNCTIONS OF TWO RANDOM VARIABLES...

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