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Unformatted text preview: 0 M\L\L?1M EEBOQ Midterm #2
MWF 3:30—4:20 PM, Prof. Gelfand Instructions: 0 There are 10 true~false problems (5 pts'each) and 2 workout problems (25 pts
each). Do all problems 9 You must Show work to receive any credit on workout 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: 0 Binomial: gm (1:) 2 (Z)pk (1  mnj, k 2 0,. ,n, Where p is average arrival rate. 7? 2
np,a%{2np(l—p).Also(Z—) W71 ":k),,0! 21. o Geometric: pT(t) 2p(1 —p)t_1,t= 1:29“7'T: ,1) 0:3“ “13—2 0 Negative Binomial order k: pT (t) 2 (1:11)pk(1 210)th ,t = 19, k+17 ~   a' T : f.) 0% "—— Mir):
k — _ . Poisson: pK (k) = 99—151, k = 0, 1, . . . ,Where /\ is the average arrival rate. K 2 At, a} 2 At A
2. o Exponential: fT (t) = Ae—Xt, t Z a. T = i7 0 ll QM
V k— H __
o Erlang orderszT(t)2§:LZc%,t >0 T=§ ‘7‘2T::\]§§
0 Uniform: fX (£102 ﬁml 2— = (142$: 0% = (121;)?
0 Gaussian: fX (x): ﬁ::P<i.(( if; ”XX—)2)
27TO'2X XlWJflz ex
c Jointly Gaussian: fXY (cc, y): W pi—Z &{~1 :X r) (($—
X,Y 1 ax
2p —2
052:1(XN7D
0 Conditionally Gaussian: fxly (ﬂy) — 2:02 6Xp( 2 ($2.33)!) 2) where leY ~— X +
X]Y M (y Y) UXLY — 0X (1 — PXY) o Leibnitz rule: dt df 5(6)) f(:1:, t) dcc2 — f(b (t) ,t) 1%? ~ f (a (t) , t) %9+ [(78 {affgﬁldm (1) function. '1) (cc): ffoo 12” exp (— —~) 03:13 Questions 1 — 10 are truefalse 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). T 1.
T T 6. 10. Ho HS ' 3. Bi’mww" i:
F
F 9.
T Consider a Bernoulli process with average arrival rate of p. Then the probability that there
are k arrivals at time t (k 2 0,1,. ..,t) is (;)pk(1 — p)t“k. . Consider a Poisson process with average arrival rate of A. Then the probability that there are k _
k arrivals at time t (It 2 0,1,..., and t 2 0) is W . Let X 2 acose and Y 2 asinG where 9 is a uniform random variable between 0 and 27r. Then X and Y are jointly continuous random variables. . Let X and Y be random variables with Cov[X, Y] 7g 0. Then X and Y are correlated but may be independent. . Let X and Y be random variables with fxy (1;, y) 2 ~21; exp (~%(a:2 + 742)) Then X and Y are independent. Let X and Y be jointly continuous random variables. Then Pr (X E A) 2 fA ffooo fxy (m, y) dydm. . Let X and Y be independent continuous random variables.Then E[g (X) h (Y)] 2 g (X) h (Y) . . Let X and Y be jointly continuous random variables, and U 2 2X + Y and V 2 2X . Then fU,V(’M, 71) = far/(53% 2 vi Let X and Y be independent continuous random variables and Z 2 X + 2Y. Then fz (z) 2
ffooo fX (53) fr (235”) dw. A signal is transmitted in noise, and the received signal is Y 2 s + N, where s 2 1 is the transmitted signal and the noise N is Gaussian with zero—mean and unit variance. Then
Pr[Y > 0] 2 @(1). Y‘,V. ’ 1‘ Voi‘gsoh ”V«
_ 1. , ‘t 2 C C ’0
3. l’v(xt*\(te a) v\ but Arra( «1%) ’x H?) A ) q. Covv{&,odro,d ’
9 i3 («(3513 XX Mu‘n‘u y‘j Jg’ql‘mH‘a 01MM9§HDW~ mHvk Jolt“! (5'39xdh‘5’0‘11 $1959 aw; PrCXthl=S [317((739L’X. , «’DO ”Ln
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§}(‘c\ ¥X‘* gubb 3:9“ 5} Mug (2 Mix  3035(0)” ch 1 \z oklx W RAY): “CSWMHWLNVSM Mun—n) = 0’32“; 3 §C\) Questions 11 and 12 are workaout problems (25 pts each). You must Show work to
receive credit. 11. Let X and Y be independent uniform random variables, each between 0 and I. Let
U :: X, V : X + Y (3.) Find fV (v) and sketch it.
(b) Find ny (u, v) and Show the region in the (u, 1)) plane where it is nonzero.
(c) Find Pr (2X < Y). 0° C): I Mag
(0” SY Ur): ¥X ¥¥Y CV) ‘2 $00 ¥x(r¥g\,(vw¥)°\’¥ $74 7‘ L
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tgvgz Cc) 12. Let X and Y be independent Gaussian random variables with mean zero and variances 0% and (7%,, respectively. Let
Z 2: aX + bY Where a, b are non—zero constants. You must express your answers in terms of 0% ,0?“ a and b and known functions (no unevaiuated integrels) (a) Find E[Z] , Var [Z], C0v[Z,X]
(b) Find E[ZX : as] and Var [ZIX = w]
(c) Find f2 (,2) and F3 (2).
(d) Find fZlX (2m) and FZ1X (zlw)
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 Fall '08
 GELFAND

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