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Unformatted text preview: ECE 302, Midterm #1
7:008:00pm 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 subquestions 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 1D 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
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“(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%
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LO l 76 at 76 t X” L 006 Question 5: [21%] Throw two unfair 6faced 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 > 0.25X < 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
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 Spring '08
 GELFAND
 Probability theory, A Closed Book, PierreSimon Laplace

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