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MT2SOL

# MT2SOL - ECE 302 Midterm#2 6:30—7:30pm Thursday March 5...

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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% A § 2"““7 z i i HE“. «E 4 \ f». \$3 1 WK 9 4: X “x 4~A a 3 3 I‘d Question 2: [18%] Let X be a Bernoulli random variable with p = 1/3. Let Y be a 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 H F .53 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 w “Miami imwmwo, . 4"; o: :3 M Y? 5/1.,» 7' i: .M x 3" m WWW Ii ‘1 “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 i . R m Ffmﬁ or tile m n (a?) r F V. iiiiii 7 r ‘ 1 i ] “MO E: A €51 r; a Micki: v ’ o Cm. w @-§0> “2‘; CD z ' “IV W W S‘\ Q‘JM _ “W’Qs; S Co‘m CAD"? mm] WWO’V'K “:1 exam-l ( W t Do (be . ,4memmmmnvwwm_ N 4 is? \‘3 Qd k": G» Céiwv’ﬁ“ {gm mg L '«mwawwmﬁwﬂv‘ ...
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MT2SOL - ECE 302 Midterm#2 6:30—7:30pm Thursday March 5...

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