100AHW1 - D )? What is the relationship between Pr( D ) and...

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STAT 100A HWI Due next Wed Note (1) If all the outcomes are equally likely, then Pr( A ) = | A | / | Ω | . (2) Conditional probability Pr( A | B ) = Pr( A B ) / Pr( B ). Problem 1: Suppose we flip a fair coin 4 times independently. (1) What is the sample space? (2) What is the set that corresponds to the event that the number of heads is 2? What is its probability? (3) Let Z i = 1 if the i -th flip is head, and Z i = 0 otherwise, for i = 1 , 2 , 3 , 4. Let X be the number of heads. Express X in terms of Z i . (4) What is the probability distribution of X ? That is, what is Pr( X = k ) for k = 0 , 1 , 2 , 3 , 4? Problem 2: Suppose we roll a fair die twice independently. Let X and Y be the two numbers we get. (1) What is the sample space? Let A be the event that X > 4, and B be the event that Y > 4. What are Pr( A ), Pr( B )? (2) Let C be the event that min( X,Y ) > 4? What is Pr( C )? What is the relationship between Pr( C ) and Pr( A ), Pr( B )? (3) Let D be the event that max( X,Y ) > 4? What is Pr(
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Unformatted text preview: D )? What is the relationship between Pr( D ) and Pr( A ), Pr( B ), Pr( C )? (4) If I tell you that X > 4, what is the probability that X = 6? (5) Let Z = X + Y , what is the probability distribution of Z ? That is, what is Pr( Z = k ), where k goes through all the possible values that Z can take? Problem 3: Suppose we draw two random numbers uniformly and independently from [0, 1]. Let X and Y be the two numbers. (1) What is Pr( . 1 X . 4)? In general, for A [0 , 1], what is Pr( X A )? (2) What is Pr( X + Y > 1 . 5)? (3) Let A be X < . 3, and let B be Y > . 6, what is Pr( A B )? What is the relationship between Pr( A B ) and Pr( A ), Pr( B )? (4) What is Pr( X 2 + Y 2 > 1)? (5) Suppose I tell you X + Y > 1, what is the probability that X > 1 / 2? 1...
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