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sMT1SOL_07S

# sMT1SOL_07S - ECE 302 Midterm#1 7:00-8:00pm Thursday Jan 25...

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Unformatted text preview: ECE 302, Midterm #1 7:00-8:00pm Thursday, Jan. 25, 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 7 questions. You have one hour to complete it. I will suggest not spending too much time on a single question, and work on those you know how to solve. . The sub-questions of a given question are listed from the easiest to the hardest. The best strategy may be to ﬁnish only the sub—questions you know exactly how to solve. . There are a total of 15 pages in the exam booklet. Use the back of eachpage for rough work. . Neither calculators nor help sheets are allowed. . Read through all of the problems ﬁrst, and consult with the TA during the ﬁrst 15 minutes. After that, no questions should be asked unless under special circum- stances, which is at TA’s discretion. You can also get a feel for how long each question might take after browsing through the entire question set. Good luck! [Liﬁfﬁ A Name: Student ID: E—mail: Signature: Question 1: [7%] Suppose ak is a series such that Question 2: [15%] Deﬁne a 1—D function fx(\$) as follows. if a: E [0, 1] if a: 6 (1, 2] otherwise fx(\$) = 0 who ooh—- Another function can be deﬁned based on the integral of f X03) as follows: Fe) = mods. Hint: Solve the ﬁrst two sub—questions ﬁrst and go back to the third sub—question if you have time. 1. [5%] Find the value of F(—1). (Hint: Do not be scared by this expression. This question is no different than asking you to compute the value of fX(s)ds.) 2. [5%] Find the value of F(0.5) 3. [5%] Find the value of assuming a: E (1, 2]. 1r Fer): ﬁred/3 : 0 ds :0 -00 Question 3: [15%] Deﬁne a 1-D function gx(x) as follows. gxim) = {S 1. [7.5%] Compute fwo:_oo xgx(:c)d\$. '33 ifm>0 otherwise 2. [7.5%] Compute the bilateral Laplace transform of gX(\$). Hint 1: The bilateral Laplace transform of any function f is deﬁned as 14(3) = :e—“f(x)da:. Hint 2: You can safely assume {3| < 1 during your computation. f?“ Kgxmﬂvr 79-420 t [:9 0(6de '90 1.- -[O w :z’ 2 L <s>~ m ~W ' 3 " g e gxcxﬁtx ~00 2 gm Q‘Me'x cfX O z [ ~Cs+mc ' m “(3%” O Hint: Solve Questions 5—7 ﬁrst and come back to this question if you have time. Question .4: [10%] Deﬁne a 2—D function f(m,y) as follows. 190 ifme 0,1 andyE 0,113 my) = / _ l. l l l 0 otherw1se 1. [10%] Compute the value of the following 2—Climensional integral. 0.75 0.5 / / f (x, yldwdy y=—oo I=—OO Yqu OS 712% @4745“ 9H)" kw ' 7c _ 0&5 MS ‘ v gm imgl 6% 05% r; 05 \x i— ate Oi LO l 76 at 76 t X” L 006 Question 5: [21%] Throw two unfair 6-faced dices and let X and Y denote the outcomes of each dice. We know that there is some invisible magnetic force between these two dices so that X will never be the same as Y. (Namely, the outcomes of these two dices will never be identical.) All other outcomes occur equally likely. 1. 3% What is the deﬁnition of “sample space”? 2. 3% What is the sample space in this experiment? 3. _4% What is the probability weight you would like to assign to each outcome of the sample space? 4. 4% What is the probability that X 2 + Y is no larger than 10? 5. 4% What is the P(Y g 2lX2 + Y s .10)? 6. [3%1 What is the P(X = 4ch2 + Y s 10)? / < The Qample Space l3 " e. Collecﬁm 61/ => PC><=4 ( X? \( 3(0)::0 Question 6: [8%] Consider an unfair 3—faced dice and each face has 4, 5, or 6 dots respec— tively. Throw this dice once and let X denote the number of dots that is facing up. Let A denote the event X S 5 and B denote the event X Z 5. Suppose we also know that (3) (4) Question 7: [19%] A real number X is randomly drawn from [0, 1]. Answer the following question. 1. [2%] What is the sample space in this experiment? 2. [3%] How to specify a weight assignment for a discrete sample space? How to specify a weight assignment for a continuous sample space? 3. [3%] What is the common equation that the total sum of any weight assignment should satisfy? 4. [3%] The probability weight assignment in this question is speciﬁed by fX(x):{c-:c ifacE[0,1]1 0 otherwise for some unknown coefﬁcient 0. What is the value of 0 should be? Hint: Use the answer of the previous question. 5. [4%] What is the probability that P(X < 0.5)? 6. [4%] What is the conditional probability that P(X > 0.25|X < 0.5)? /‘ S: ‘2 ~ :I. S QC: [1 ' 1* Ar - r em m a Sac/i ear ‘ A [D l\,/ “five W6 ]% , OLA m “(he Sum :1 1": Q Com/«e, Suck “that ﬁle weight is The. Weak UMOQYW& ﬁes Cow/e "The “(fated area S‘Aoudoi be 2D(X>Otl§ and X<o£) 0‘5— L'O‘S = S 2X0Q7< = x( 09—5 0625 r: x? [6 % PCXNWH X035) «3‘ 1:: (6 .._ “L _. 4, ...
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sMT1SOL_07S - ECE 302 Midterm#1 7:00-8:00pm Thursday Jan 25...

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