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4703-10-Fall-h1

4703-10-Fall-h1 - IEOR 4703 HMWK 1 Professor Sigman Let U...

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IEOR 4703, HMWK 1, Professor Sigman Let U be uniformly distributed on the interval (0 , 1); P ( U x ) = x, x (0 , 1). We as- sume that your computer can sequentially generate such U independently upon demand. 1. Let X = ( b - a ) U + a . Show that X is uniformly distributed on the interval ( a, b ); that is, show that P ( X x ) = ( x - a ) / ( b - a ) , x ( a, b ). Thus, if your computer can generate a U , then it can in fact generate any rv X that is uniformly distributed on the interval ( a, b ) for any desired a and b . 2. Continuation: X has cdf given by F ( x ) = x - a b - a if x ( a, b ) , 1 if x b, 0 if x a. Show that in fact setting X = ( b - a ) U + a is exactly the inverse transform method. 3. Let V = 1 - U . Show that V is also uniformly distributed on the interval (0 , 1). (Use this fact whenever using the inverse transform method if you have a chance to replace 1 - U with U in an algorithm.) 4. Pareto distribution (power tail): Consider the cdf F ( x ) = ( 1 - 1 /x 2 if x 1 , 0 if 0 x < 1 .

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