Mathematical Statistics Quiz#1-SOL-2008

Mathematical Statistics Quiz#1-SOL-2008 - Mathematical...

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Unformatted text preview: Mathematical Statistics Solutions to Quiz #1 4/10/2008 1. (1) Suppose that we have r boxes. Balls are placed at random one at a time into the boxes until, for the first time, some box has two balls. Find the probability that this occurs with the nth ball. SOL: (2) Consider the problem of matching n objects, and let i and r denote distinct specified positions. (i) What is the probability that a match occurs at position i and no match occurs at position r? (ii) Given that there is no match at position r what is the probability of a match in position i? SOL: (i) (ii) 2. Prove the following statements. (1) If A, B, and C are three events such that P A B C P C|A B P C|B , then P A|B C P A|B . (2) P A B PA PB 1. SOL: (1) P C A PA B B PA B C PA B 0 and P C|B B PB C PB PB C PB C PA C PA BPB C PB CPB Therefore, P A B (2) P A B PA PA B PA B C PB C P A|B . PB PA PA PB B and P A PA B PA B 1 PB 1 3. Consider the trivariate probability density f x ,x ,x x xe 0 , for 0 x 1, 0 x 1, 0 otherwise x , (1) Find the marginal density of X and X and the marginal density of X alone. SOL: f , x ,x x x x xe xe e dx dx d x dx x x e 0 x 1, 0 x f x , 0 x 1 (2) Find the conditional expectation of X given X SOL: f x |X x ,X x x 2 x xe 1 e 2 0, 2 and X ,0 x 2 2. 1 fx X 1 ,X 3 1 3x 11 32 6x ,0 5 x 1 otherwise 1 x 5 2 x 5 x x 1 0 3 5 EX X 1 ,X 3 2 x 6x dx 5 (3) Are the three random variables independent? SOL: X , X and X are not mutually independent. X and X , X and X are independent X and X are not independent. 4. (1) Suppose that X is a random variable for which the moment generating function is as follows: M t e e e for ∞ ∞. Find the probability distribution of X. SOL: P X 2 SOL: M t ! 1 , PX 1 t 4 t EX t , PX t n! 8 ∑ . EX ! EX 1 5. Let X be a nonnegative integer‐valued random variable. Suppose that the series ∑ P X x converges. Show that E X ∑ P X x . (Hint: x ∑ 1 and write E X as a double summation. Since the series converges, you can interchange the order of summation if you want to.) ∑ ∑ 1PX x ∑∑ ∑ xP X x SOL: E X PX x Interchange the order of summation, we have ∑ ∑ PX x ∑ PX y ∑ PX x . 6. Suppose that X and Y are two independent random variables with the same probability distribution f x p 1 p , x 0, 1,2, . (1) Find P Y X . ∑ ∑ P X x, Y y ∑ ∑ p1 p p1 p SOL: P Y X ∑ p1 p (2) Find P X Y z . ∑ P X x, Y z x SOL: P X Y z 1 zp 1 p ∑ p 1 p (3) Find P Y y|X Y z SOL: P Y y|X Y z PY ,X Y PX Y ∑ p1 p p1 p 7. (1) If X is a random variable with the mean µ and variance σ , then for any value k 0, P |X µ| k . SOL: See text. (2) Let X , X , , X be n independent random variables with the same mean µ and variance σ and X be the sample mean of X , X , µ| k 0 as n ∞. P |X SOL: By (1) P |X µ| k VX , X . Prove that 0 as n ∞ . 8. Let T ∑N X , where N is a random variable with a finite expectation and the X are the random variables that are independent of N and have finite mean E X . (1) Show that the moment generating function of T is MY t (Hint: Use the iterative conditional expectation). (2) Use the result in (1) to show that E T E N E X and E N V X . VT EX VN SOL: (1) Conditioning on N E e ∑N X E MX t N |N n Ee ∑ X MX t and thus MY t (2) E T EEe MY 0 ∑N X |N n N N E MX t MX 0 MX t N . EXEN N E N MX 0 1 MX t ENN E NE X N MX t NE X MY t ENN And so E T V T MY 0 MX t 1 EX EX EN EN EN EX EX ENVX EX EN ET ET ENVX EX EN ENVX EX EN ENVX E N E X EX EN EX EN EN EX VN ...
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