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EE302 Midterm #1 Section 2, TR 12 — 1:15, Prof. Gelfand Instructions: 0 There are 10 truefalse problems (5 pts each) and 2 workout problems (25 pts
each). Do all problems 0 You must ShOW work to receive any credit on work—out problems.
0 Calculators but not laptops are allowed
0 Cheating will result in failure of the course. Do not Cheat! 0 Put your name on every page of the exam and turn in everything when time is
called. Useful formula (the mathematical formula in Appendix A of the text is also appended
to the end of the exam): 0 Binomial:I Pr (X : k)_= (71:)pl” (1 — p)n*k, k 2 0, . . .,n, where p is probability of arrival.
o Exponential: fx (r) : APT)”, m 2 0. X : f /\ /\.r “1’16"” 0 Erlang order n: fx (”5) = (#1)! ,m > 0. Y 2 >43 0 Uniform: fX (1') = ﬁ, (1 < a: < b. (27f)  (D function: <1) (:L') = ffoo L exp (_%2) dr 27r LODd 0 Gaussian: fX (:15) = l
V27702 exp (— Questions 1  10 are true—false problems (5 pts each). Label each statement true or
false to the left of the problem number. (note: if statement is not always true, then
it is false). E1KJ\WW°HOV‘S ‘ F 1.
‘2 2.
T 3
T 4.
F 5
1 6.
F 7.
1— 8
T 9.
T 10.
t. A — Pr
. P (AURUL) — PHAH
”l r + Drmw— PrCU" PrCEnc) ' Pr((m\!‘$)u(nnc)) iMA—B): (9,(PmTZ): prtmmrui) : Let A,B and C be sets. Then AU (Em C) = (AﬂB) U (Am 0’) Let A, B and C be events. Then Pr (A U B U C’) : Pr (A) + Pr (B) + Pr (C) — Pr (A H B) —
Pr(AﬂC)—Pr(BﬂC). . Let A and B be independent events Then Pr (A — B) = Pr (A) (1 * Pr (B)). Let A and B be events (Pr (A) > 0). Then Pr (BlA) : Pr (AlB) Pr (B) / Pr (A) . . Let X be a random variable. Then Pr (X < .72) : FX (33) . Let X be a continuous random variable. Then Pr (X < 3:) = ffoo fX (x) dx. Let X be a discrete random variable which takes a ﬁnite number of values, and Y = g (X)
Where g(:1:) is not one—to—one (i.e.7 g(ml) 5A 9(332) for some 331 9E 132). Then Y is a discrete
random variable which takes fewer number of values then X. . Let X be a continuous random variable and Y : g(X), where g(:z:) is a smooth strictly increasing function (i.e., 9(321) < g(mg) for all :51 < 3:2). Then fy (y) : f); (cc) g—j when
y 6 range (9) . Let X be a random variable and a and b be constants. The Var [aX + b] 2 a2 Var [X] . Let X be a random variable. Then Var (X) Z 0, and is equal to zero if and only if X is a
constant. u (Mo = (Av8)n(bruc) (Bur) ' Pr Centauo) ‘ Prlﬁ)
—“ PHHH 9MB) r 9r“)  Pr (gm) “(PM/5mm + prcp‘ne) , yrcnnfénc3) Prlxcx)’F(7~) + l’rix‘ix) Ur ?r(7K«oc)>e PM“ WW“ it (”cud—5 MAWSS ‘3'“) E»), )Q: endxx” Amwgk W5» rub“; rtdz (bud ”‘3 bwhqﬂ CHM/2 ?h(A)(\'Pr(§)) he 3:33 0d" MM Cl HM “‘4 Wu): K com tabt J t * w $29.me 6% Questions 11 and 12 are work—out problems (25 pts each). You must show work to
receive credit. 11. Do either A or B. A. Leif Wrov‘ Srw Mzsswge) u; M“ 8— FWVSWCS Computer A sends a message to computer B simultaneously over two unreliable tele—
phones. Computer B can detect when errors have occured in either line. Let the prob—
ability of message transmission error in line 1 and line 2 be Q1 and qg, respectively.
Computer B requests retransmission until it receives an error—free message on at least
one of the lines. Errors in different lines and/ or at different retransmission times are all
independent. Find the probability that at least n transmissions are required. (Hint: the
probability that at least 77. transmissions are required is the same as the probability that
a certain event does not occur in n trials for a particular sequence of Bernoulli trials). . A multichannel microwave link is to provide telephone communications to a remote community having n subscribers, each of which is active independently with probability
p. The link can provide up to k seperate channels, each of which can accomodate a single
subscriber. Find the probability that the link can accomodate all of the 71 subscribers, in
terms of p, k: and n. (Hint: the probability that the link can accomodate all 77. subscribers
is just the probability that the link can provide a channel for each of the subset of n
subscribers that are active.) — 4. )k Xi‘— l—Irowsvw‘35f°”5 at: h A— Lwcwauvtﬁ S‘irs’r
‘ 3' N: it ewuw 51mg WISSD‘S‘W {tIW( ‘ ’n/W'V“ QrQK>/°"\ ’ 3.. k v ra 0.8
c 25 WW 12. Let X be a random variable and Y = g (X) , Where 2 .
a: w>—1
g(m)={1’ mE—lj Suppose X is a uniform random variable between —2 and 2. (a) Find and plot the probability density function and probability distribution function of
Y. (b) Find the probability that Y Z 1.
(c) Find the mean of Y
((1) Find the variance of Y. .39”. it;
. In *7. 1‘; 'x ”L ‘L
_ 3L .) + 9t») (‘0 91W): VrC‘hﬂ I? C \d:c\ 0,334!
(g (a): ¥X(*I)lo:—:gl " gxm“) *3
*3 2
d.
Vrf‘f:;\: Pr()ge)_5bo¥$(m II~ ‘ , am; ._. .4
we: we, «343) us" as
5M” g“ I; a,” Euro J, rem 'do
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 Fall '08
 GELFAND

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