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Main points of this exam paper are: Probability, Space, Corresponds, Random Experiment, Replacement, Perfectly Shuffled, Uniformly Distributed
Typology: Exams
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Department of EECS - University of California at Berkeley EECS126 - Probability and Random Processes - Fall 2003 Midterm No. 1: 10/3/
Name and SID:
There are six questions. Answer on these sheets. Show your work. Good luck.
Question 1 (15%). Is it true that
P (A ∩ B ∩ C) = P [A | B]P [B | C]P (C)?
If true, provide a proof; if false, provide a counterexample.
Question 2 (15%). Describe the probability space {Ω, F, P } that corresponds to the random experiment “picking five cards without replacement from a perfectly shuffled 52-card deck.”
Question 4 (15%). Let (X, Y ) be the coordinates of a point picked randomly and uniformly in [0, 1]^2. Calculate P [X + 2Y ≤ 1 | 2 X + Y ≤ 1].
Question 5 (15%). Let X be a random variable that is exponentially distributed with mean 1. Calculate P [X ∈ [1, 4] | X ∈ [3, 5]].
