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Course: STAT 3092006, Fall 2009
School: Wisconsin
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Word Count: 356

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DISCUSSION STAT309 3 TA: Min Niu E-mail: niu@stat.wisc.edu Office Hour: 1:00pm-2:00pm Tuesday &amp; 5:30pm-6:30pm Thursday Location:Room B248 MSC Homepage:http://www.stat.wisc.edu/ niu/stat309.html 1. Basic Concepts (a) Uniform probability on finite spaces: The uniform probability distribution on a finite sample space S satisfies P (A) = |A|/|S|. (b) Combinatorial Principles : Multiplication principle: |S| =...

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DISCUSSION STAT309 3 TA: Min Niu E-mail: niu@stat.wisc.edu Office Hour: 1:00pm-2:00pm Tuesday & 5:30pm-6:30pm Thursday Location:Room B248 MSC Homepage:http://www.stat.wisc.edu/ niu/stat309.html 1. Basic Concepts (a) Uniform probability on finite spaces: The uniform probability distribution on a finite sample space S satisfies P (A) = |A|/|S|. (b) Combinatorial Principles : Multiplication principle: |S| = |S1 |...|Sk | Permution: n! = n (n - 1) 2 1 n n! Binominial coefficient: = k!(n-k)! k n Multinomial coefficient: k 1 k2 kl (c) Conditional Probability: (AB) P (A|B) = P P (B) Let A1 , A2 , ... be events that form a partition of the sample space S, let B be any event. Then P (B) = P (A1 )P (B|A1 ) + P (A2 )P (B|A2 ) + ... P (A) P (A|B) = P (B) P (B|A) A and B are independent if P (A B) = P (A)P (B) = n! k1 !k2 !kl ! 2. Examples (1) Consider the following three games. Game 1: Throw a single die 6 times. If at least once, your score is 1 then you win the game. Game 2: Throw a single die 12 times. If at twice, least your score is 1 then you win the game. Game 3: Throw a single die 18 times. If at least thrice, your score is 1 then you win the game. In which game do you have the highest probability of winning? (2) Suppose we roll ten fair six-sided dice. What is the probability that there are exactly two 2's showing, and exactly three 3's showing? (3) Suppose we pick a card at random from an ordinary 52-card deck, and also flip ten fair coins. What is the probability that the number of heads equals the value of the card (where we count jacks, queens, and kings as ten, and count aces as one)? (4)Suppose the probab...

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Wisconsin - STAT - 3092006
STAT309 DISCUSSION 8 TA: Min Niu E-mail: niu@stat.wisc.edu Office Hour: 1:00pm-2:00pm Tuesday &amp; 5:30pm-6:30pm Thursday Location:Room B248 MSC Homepage:http:/www.stat.wisc.edu/ niu/stat309.html 1. Basic Concepts If X and Y are discrete,then the condit
Wisconsin - STAT - 3092006
STAT309 DISCUSSION 9 TA: Min Niu E-mail: niu@stat.wisc.edu Office Hour: 1:00pm-2:00pm Tuesday &amp; 5:30pm-6:30pm Thursday Location:Room B248 MSC Homepage:http:/www.stat.wisc.edu/ niu/stat309.html 1. Basic Concepts If X is discrete, then E(X) =xxP (X
Wisconsin - STAT - 3092006
STAT309 DISCUSSION 10 TA: Min Niu E-mail: niu@stat.wisc.edu Office Hour: 1:00pm-2:00pm Tuesday &amp; 5:30pm-6:30pm Thursday Location:Room B248 MSC Homepage:http:/www.stat.wisc.edu/ niu/stat309.html 1. Basic Concepts The variance of a random variable X me
Wisconsin - STAT - 3092006
STAT309 DISCUSSION 11 TA: Min Niu E-mail: niu@stat.wisc.edu Office Hour: 1:00pm-2:00pm Tuesday &amp; 5:30pm-6:30pm Thursday Location:Room B248 MSC Homepage:http:/www.stat.wisc.edu/ niu/stat309.html 1. Basic Concepts Let X and Y be discrete random variabl
Wisconsin - STAT - 3092006
STAT309 DISCUSSION 12 TA: Min Niu E-mail: niu@stat.wisc.edu Office Hour: 1:00pm-2:00pm Tuesday &amp; 5:30pm-6:30pm Thursday Location:Room B248 MSC Homepage:http:/www.stat.wisc.edu/ niu/stat309.html 1. Basic Concepts A sampling distribution is the distrib
Wisconsin - STAT - 3092006
STAT309 SOLUTION 6 2.6.2 Let h(x) = cx+d. Then Y = h(X) and h is strictly decreasing, so fY (y) = fX (h-1 (y)/|h (h-1 (y)| = fX (y - d)/c)/|c|, which equals 1/(R - L)|c| = 1/(cL - cR) for L (y - d)/c R, i.e., cR + d y cL + d, otherwise equals 0.
Wisconsin - STAT - 3092006
STAT309 SOLUTION 9 3.3.2 (a) E(X)=(5)(1/7)+(5)(1/7)+(5)(1/7)+(8)(3/7)+(8)(1/7)=47/7. Also, E(Y )=(0)(1/7)+(3)(1/7)+(4)(1/7)+(0)(3/7)+(4)(1/7)=11/7. (b) E(XY )=(5)(0)(1/7)+(5)(3)(1/7)+(5)(4)(1/7)+(8)(0)(3/7)+(8)(4)(1/7)=67/7. Then Cov(X, Y ) = E(XY )
Wisconsin - STAT - 3092006
STAT309 SOLUTION 11 4.1.1 P (Y3 = 1)=(1/2)(1/2)(1/2)=1/8 P (Y3 = 2)=(1/4)(1/4)(1/4)=1/64 P (Y3 = 3)=(1/4)(1/4)(1/4)=1/64 P (Y3 = 21/3 )=(1/2)(1/2)(1/4)+(1/2)(1/4)(1/2)+(1/4)(1/2)(1/2)=3/16 P (Y3 = 31/3 )=(1/2)(1/2)(1/4)+(1/2)(1/4)(1/2)+(1/4)(1/2)(1/2
Wisconsin - STAT - 3092006
STAT309 SOLUTION 13 4.4.1 Here limn P (Xn = i) = 1/3 = P (X = i) for i=1,2,3, so limn P (Xn x) = P (X x) for all x, so Xn X in distribution. 4.4.4 For 0 &lt; w &lt; 1, P (Wn w) = 0 (1 + x/n)/(1 + 1/2n)dx = (w + w2 /2n)/(1 + 1/2n) w as n . Also, P (W
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