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100Amidterm

# 100Amidterm - STAT 100A MIDTERM EXAM Solution Problem...

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STAT 100A MIDTERM EXAM Solution Problem 1: Suppose we generate two independent random variables X and Y uniformly over [0 , 1]. (1) (4 points) Calculate P ( X 2 + Y 2 1). A: Let Ω be the unit square [0 , 1] 2 , and let A be the event that X 2 + Y 2 1, then P ( A ) = | A | / | Ω | = π/ 4. (2) (6 points) Calculate P ( X > 1 / 2 | X + Y < 1). A: Let A be the event that X > 1 / 2, and let B be the event that X + Y < 1, then P ( A | B ) = P ( A B ) /P ( B ) = (1 / 8) / (1 / 2) = 1 / 4. Problem 2: Suppose at any moment, the probability that there is fire in a particular building is α . Given there is fire, the probability that the fire alarm is heard is β . Given there is no fire, the probability that the fire alarm is heard is γ . (1) (5 points) At any moment, what is the probability that we hear the fire alarm? A: P (alarm) = P (fire) P (alarm | fire) + P (nofire) P (alarm | nofire) = αβ + (1 - α ) γ . (2) (5 points) If we hear the fire alarm, then what is the probability that there is actually a fire? A: P (fire | alarm) = P (fire alarm) /P (alarm) = P (fire) P (alarm | fire) /P (alarm) = αβ/ ( αβ +(1 - α ) γ ).

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100Amidterm - STAT 100A MIDTERM EXAM Solution Problem...

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