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HW03_2011

# HW03_2011 - X tends to be bigger than Y 3.6 Suppose X = X 1...

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PROF Y. HONG FALL 2011 ECONOMICS 6190 PROBLEM SET # 3 3.1 Can two independent events A and B be mutually exclusive? Can two mutually exclusive events A and B be independent? Give your reasoning. 3.2. Give an example to show that P ( A \ B \ C ) = P ( A ) P ( B ) P ( C ) cannot guarantee P ( A \ B ) = P ( A ) P ( B ) . 3.3. Seven balls are distributed randomly into seven cells. Let X i = the number of cells containing exactly i balls. What is the probability distribution of X 3 ? (That is, °nd P ( X = x ) for every possible x .) 3.4. [C&B, # 1.47, p.43] Prove that the following functions are cumulative distribution func- tions(CDF±s): (1) 1 2 + 1 ° tan ° 1 ( x ) ; x 2 ( °1 ; + 1 ); (2) (1 + e ° x ) ° 1 ; x 2 ( °1 ; + 1 ); (3) e ° e ° x ; x 2 ( °1 ; + 1 ); (4) 1 ° e ° x ; x 2 (0 ; + 1 ) : 0 ; x ± 0 3.5. A CDF F X is stochastically greater than a CDF F Y if F X ( t ) ± F Y ( t ) for all t and F X ( t ) < F Y ( t ) for some t . Prove that if X ² F X and Y ² F Y , then P ( X > t ) ³ P ( Y > t ) for every t P ( X > t ) > P ( Y > t ) for some t; that is,

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Unformatted text preview: X tends to be bigger than Y . 3.6. Suppose X = X 1 with probability p and X = X 2 with probability 1 & p; where p 2 (0 ; 1) ; X 1 and X 2 are random variables with CDF²s F 1 ( x ) and F 2 ( x ) respectively. Find the CDF of X: 1 3.7. An investment &rm o/ers its customers municipal bonds that mature after varying numbers of years. Given that the cumulative distribution of T; the number of years to maturity for a randomly selected bond is F ( t ) = 8 > > > > > > < > > > > > > : t < 1 ; 1 4 1 & t < 3 ; 1 2 3 & t < 5 ; 3 4 5 & t < 7 ; 1 t ± 7 : Find (1) P ( T = 5); (2) P ( T > 3); and (3) P (1 : 4 < T < 6) : Give your reasoning. 2...
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HW03_2011 - X tends to be bigger than Y 3.6 Suppose X = X 1...

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