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100AHW1S

# 100AHW1S - STAT 100A HWI Solution Problem 1 Suppose we ip a...

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STAT 100A HWI Solution Problem 1: Suppose we flip a fair coin 4 times independently. (1) What is the sample space? A: The sample space Ω consists of all the 2 4 = 16 sequences of heads and tails. (2) What is the set that corresponds to the event that the number of heads is 2? What is its probability? A: The event is { HHTT, HTHT, HTTH, THHT, THTH, TTHH } . The probability is 6 / 16 = 3 / 8. (3) Let Z i = 1 if the i -th flip is head, and Z i = 0 otherwise, for i = 1 , 2 , 3 , 4. A: Let X be the number of heads. Express X in terms of Z i . A: X = X 1 + X 2 + X 3 + X 4 . (4) What is the probability distribution of X ? That is, what is P ( X = k ) for k = 0 , 1 , 2 , 3 , 4? A: Using the same method as in the answer to question (2), P ( X = 0) = 1 / 16. P ( X = 1) = 1 / 4. P ( X = 2) = 3 / 8. P ( X = 3) = 1 / 4. P ( X = 4) = 1 / 16. 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 P ( A ), P ( B )? A: The sample space Ω consists of the 36 pairs of numbers. P ( A ) = P ( B ) = 2 / 6 = 1 / 3.

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100AHW1S - STAT 100A HWI Solution Problem 1 Suppose we ip a...

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