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Unformatted text preview: ECE 302, Midterm #2
6:30—7:30pm Thursday, March 5, 2009, BB 170, . Enter your name, student ID number, e—mail address, and signature in the space
provided on this page, NOW! . This is a closed book exam. . This exam contains only work—out questions. You have one hour to complete it.
The students are suggested not spending too much time on a single question, and
working on those that you know how to solve. . There are 12 pages in the exam booklet. Use the back of each page for rough work.
. Neither calculators nor help sheets are allowed. . You can rip off the table in the back of the exam booklet. Name:
Student IT):
E—mail: Signature: Question I : [20%] 1. [3%] X is an exponential random variable with /\ z 2. Find E ((X — 0.5)2) [3%] X is a binomial random variable with n = 100, p = 0.1_ Find P(X : 0]X < 2). 3. [3%] X is a Poisson random variable with 04 = 1. f is a function such that 3 if ——().1 <x§ 1.5
f(:z:) = —2 if 1.5 < :1: s 1.9 (1)
0 otherwise Find E(f(X)2).
4. [5%] X is a geometric random variable with p = 1/5. Find E (2X — X 5. [6%] X is a uniform random variable with a = 1, b = 3. Find E(rnax(X, 2)) “Mr”? «v “:M 33%
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binomial random variable with n = 2 and p = 1/4. Suppose X and Y are independent.
Answer the following questions: 1. [3%] What is the sample space when considering jointly (X , Y)? What is the corre—
sponding weight assignment? 2. [10%] Let Z = X3Y. Find and plot the cdf 3. [5%] Find and plot the generalized pdf If you do not know the answer to the
previous question, you can assume 172(2) being as follows. 0 if 2 < 0
0.25 sin(z) if 0 g z < 1 Fz(Z)= g3: if1<z<jr (2)
7r 2 ~— 1 ifwgz [a
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Q mew“whuwmdmmmkﬁv , ,w Question 3: [7%] Consider a discrete random variable W with Cdf Fw(’w) being 0 if w < 0.5
Fw(w) : 0.01102 if 0.5 s w < gr 3 (3)
1 — sing—5) if 27r 3 w L63). :3. Find the probability P(0.5 g W < 4 or 5.5 < Question 4: [25%] Consider a random variable X such that S = (1,00), i.e., X can be
any larger—than—one real numbers (say 1.1, 7r, etc). We also know that the probability
density of X is fX(ac) = 0649” for x > 1 where c is a constant. 1. [3%] Find the 0 value. 2. [10%] Find the characteristic function <I>X(w). W [12%] Use the moment theorem to ﬁnd the mean and the variance of X. (If you do
not know the answer of the previous question, you may assume <I>X(w) = ej‘“_1(1 + $351) WM...W~..WM.M.N, #7:) z . x
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“WE. V) Question 5: [30%] Consider the following game. You choose a number a ﬁrst. Then the computer
generates a number X that is an exponential random variable with /\ = 0.5. If X > a,
then you receive a reward of a dollars. If X g a, then you receive nothing. 1. [8%] What is the expected reward of this game in terms of a? Hint: It might be
easier to ﬁrst consider a special case a = 3. In this case, your reward function is 3 ifx>3
f($):[01fx33' (4) You are interested in the expected reward E(f(X 2. [3%] If you play this game, what value of a should you choose? (If you do not know (in we the answer to the previous question, you can assume the expected reward is Zlee—2a.)
#0 A Hint: Maximize your expected reward by considering the ﬁrst order derivative.
exam » 3. [14%] Suppose an oracle tells you that the next time you play this game, X must
be greater than 2. Given that X > 2, what is the conditional expected reward of
this game in terms of a. 4. [5%] How to tell whether a RV. is discrete, or continuous, or of mixed type form its
cdf? Plot any one cdf FX that corresponds to a random variable of mixed type. I i? w i
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 Spring '08
 GELFAND
 Probability theory, CDF

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