UCSD ECE 153 Handout #17
Prof. Young-Han Kim Tuesday, November 4, 2008
Midterm Examination
(Total: 120 points)
There are 3 problems, each problem with 4 parts, each part worth 10 points.
Your answer should be as clear and readable as possible. In particular, if the answer involves
a pmf or pdf, make sure to identify the values or intervals for which the pmf or pdf is nonzero.
1. Coin with random bias.
A random variable Pis drawn uniformly from the interval [0,1]. Then a coin with bias
Pis flipped three times. Assume that the value of the bias does not change during the
sequence of tosses.
(a) What is the probability that all three flips are heads?
(b) Find the probability that the second flip is heads given that the first flip is heads.
(c) Is the second flip independent of the first flip?
(d) What is the conditional pdf of the bias Pgiven the first flip is heads?
2. Two independent uniform random variables.
Let Xand Ybe independently and uniformly drawn from the inverval [0,1].
(a) Find the pdf of U= max(X, Y ).
(b) Find the pdf of V= min(X, Y ).
(c) Find the pdf of W=U−V.
(Hint: In case you are stuck, you can start working on part (d) first.)
(d) Find the probability P{|X−Y| ≥ 1/2}.
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