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Unformatted text preview: Sothmvs EE302 Midterm #2
TR 7:30—8:45 AM, Prof. Gelfand Instructions: There are 10 truefalse problems (5 pts each) and 2 workout problems (25 pts
each). Do all problems You must show work to receive any credit on workout problems.
Calculators but not laptops are allowed
Cheating will result in failure of the course. Do not cheat! Put your name on every page of the exam and turn in everything when time is
called. Useful Formula: Binomial: pK (k) = (71:)pk (1 ~p)"_k, 19 = 0, . . .,n, Where p is average arrival rate. Y? =
np,a§{ =np(1 —p). Also (71:) = [cl—(71%,0!= 1. Geometric: pT(t) =p(l—p)t"1,t= 1,2,...,.T= i012; : 1—152.
Negative Binomial order k: pT (t) : ill)“ (1 —p)t'k ,t = k, 19+ 1,. . .,. T = g, 0% = k if? . k _ .._
Poisson: pK (k) = W, k = O, l, . . . ,where A is the average arrival rate. K = At, 0% = /\t. Exponential: fT (t) = Ae‘dt, t 2 0. T = %, 0%, = i_ k— — __
Erlangorderk:fT(t)=%,t20_T=§, 033,2}? Uniform: fx (:3) = Fl?» (1 < :c <
— 2
Gaussian: fX (m) = ﬂap (_% (mo—f) > .
X (1) function: (13(32): ”5 —1— (—%2 dx. —oo ‘57; exp 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. Let X and Y be jointly continuous random variables. Then Pr (X + Y Z 1) = ff; fffy fxy (3:, y) dmdy
T 2. Let X and Y be jointly continuous random variables, and let U = aX + bY where a and b
are positive numbers. Then fay (u, y) = if)“, (”—j’y, y)
‘2 3. Let X and Y be jointly continuous random variables, and let Z = X + Y. Then fle (zlz) =
fy (z — m) . 4. Let X and Y be jointly continuous independent random variables, and let 9 (w) and h (y) be
functions. Then E[g (X) h(Y)] = E [g (X)] E [h (Y)] 5. Let X and Y be random variables and a, b, c, d be numbers. Then Cov [aX + bY, cX + dY] =
ac Var [X] + bd Var [Y] . random variables. 7. Let X and Y be jointly Gaussian random variables. Then fXIy (xly) is a Gaussian density in T
F
T 6. Let X and Y be random variables. If iprI = 1 then X and Y cannot be jointly continuous
T
x. 8. In a signal quantization system, a uniformly distributed random variable on [~a, a] is input to a uniform quantizer with 2 levels. Then the root mean square error is f
T 9. In a digital communications system, the received signal is Y = s+N when a “1” is transmitted, and Y = N when a “0” is transmitted, where the signal 3 > O is known, the noise N
is Gaussian with mean 0 and variance 02, and the transmitted bits are equally likely and
independent of the noise. The decision on the received signal is made according to: Y > 5 say
a “1” is transmitted, and Y S gsay a “0” is transmitted. Then the probability of error is 1 — (I) (ﬁ).
F 10. In a sequence of independent coin ﬂips with an unfair coin, the probability of a head on any
[6‘1 at
ﬂip is or. Then the probability that the k — th head occurs on ﬂip t is 526371)?— (k = 1, 2, . . .,
and t: 1,2,...).
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8 3:  h A».   Questions 11 and 12 are workout problems (25 pts each). You must show work to
receive credit. 11. Let X and Y be jointly continuous random variables with pdf bra/(any) = 0, OSxSyS 1,
0, else. where c is a constant. ()  (b) Find Cov [X , Y]. Are X and Y uncorrelated? ) Find fx (9:) , fy (31) . Are X and Y independent? as?"
) SHCM33=F sxcxugycj) C9 <~ 5709‘ 5d” = 2% “nu m m m
l I I ‘ indeyen denl: . “x: 920—0«61 := \_g_ = y/ > Cc Gov) l4 éxgg _ 36 :. X3: y am cameloled ?Mo‘o\em H COthvecl. §2C73= 0 ‘50 12. A circuit component burns out on average once a day. When a component burns out, it
is immediately detected and replaced with a new component. Note that the burnout and
subsequent replacement can happen at any time of the day. Assume that the replacement
times follow a Poisson process, starting at day zero. (a) Find the pdf, mean and variance for the amount of time up to the k — th component
replacement. (b) Find the pmf, mean and variance for the number of component replacements in t days. (0) Find the conditional pdf, mean and variance for the time of the ﬁrst replacement given
that the component which started at day zero has lasted at least 1 day. 0/») The h‘me between been outs is ex\>onen\{0cl with Pmameiea A e i . ’n‘me M1 the R“ bum—out: is cm KH‘ omdexfmlanﬁ dislmwtgcj Pondmm mane, with W” K Kl _
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 Fall '08
 GELFAND

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