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prevexam2b

# prevexam2b - EE302 Midterm#2 Section 2 TR 12:00 1:15 Prof...

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Unformatted text preview: EE302 Midterm #2 Section 2, TR 12:00 - 1:15, Prof. Gelfand Instructions: There are 10 true-false problems (5 pts each) and 2 work-out problems (25 pts each). Do all problems You must ShOW work to receive any credit on work-out 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 (the mathematical formula in Appendix A of the text, including integrel tables, is also appended to the end of the exam): #— Binomial Pr (X — k:—) 2 (2)101“ (1 —p)” k .k2 0 ,n where p is probability of arrival. (Z)=m70'—1-X=”P- Geometric: Pr(X2 k)2p(1—p)k’1,k21,2=...; X2 ‘dh— Pascal order n: Pr (X— k): (:_1)PTL(1 “ 29)] n k_ — n 7H" 1 X 2 %‘ Poisson: Pr (X2 k): m—“", k 2 0, 1, . . . ,where A is the average arrival rate. 35 2 At. Exponential: fX (1:) 2 A€_)‘I, :1: 2 0, Y = i Erlang order n : fX (2:) 2 W x Z 0. 3(— Uniform. fX (3:): —— a < a: < b 1a) «217m (—% (1;?)2) Gaussian: fx (3:) 2 ‘ ' 1 l 513—} 2 L 2 2 Jomtly Gaussmn: ny (2:,y) — QWUXUYM exp 20%) ( 0X ) + ( 0X ) — 2&0”), (a: — X (I) function: (I) (It): fix)“ 2” exp (—3? ;) dz Leibnitz rule a "3&be f(a:, 15) (£132 f(b(t) t)9’_b(t>_ W (t) ﬂda am + farm/(Jodi 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). T 1. Let X be a continuous random variable with density function fT (t )2 e‘t ,t > O (0 otherwise) Then fT (tIT > a)— — e (t a), t Z a > 0 (0 otherwise). F 2. Let X and Y be discrete random variables with joint mass function pr (rhyj) . Then 4 _ .___‘1*__ . . . E [XY] _ Eli 2y]. miyij,Y(I£yy]) (when the right Side ex1sts) F 3. Let X and Y be continuous random variables with joint density function ny (3:, y) = kry, O S a: S y S l (0 elsewhere), where k is the correct normalization constant. Then X and Y are independent. T 4. If X and Y are independent random variables then so are X 2 and Y3. F 5. If X and Y are uncorrelated random variables then so are X 2 and Y3. T 6. If X and Y are jointly Gaussian uncorrelated random variables, then they are independent. T 7. The minimum number of parameters necessary to specify the density of 3 jointly Gaussian random variables is 9. T 8. If X and Y are random variables with U_2X = 021/, then X + Y and X v Y are uncorrelated. F 9. If X and Y are random variables with pr : 1, then X = Y. F; 10. Let X and Y be independent continuous random variables, and Z : (1X + bY, where a and b are positive constants. Then f2 (2): LOO M( ) fy (2'1”) du EttiaumK-NS' oanMuytx wed- VQObqueVi uwﬁk M0 {90.th and”, e' r 3 ‘s'T(i’iT>0\ : ﬂ) ,__ 3::- t C 0‘ E>R 06 e-& Laid“ 1 EL: 3 2 E «if . Pmi"*»‘33) 1‘ 33' “is L \ ~ it ) glut “M 7"5 7:. iX¥Cmgﬂit 9‘“ 1’3” M- ari, F ‘l’. (3170 an) LAY) an L d:\, u.- S- 24)“ and i/\(\l\ Q‘V‘ (XL) aw} ML) an Hutu?) Ma} metﬁgiuvn’ﬁ LLVLLur‘U/cio-i-avi (wakeﬁs (‘2 'L 1- X‘I¥I,X3,‘x..°‘%1rﬁx3,PKrXL,JD7“*3.szX3 53/. CWLXtY,X—Y): \Iuv’t‘ﬂ‘i‘vﬁrf‘ti)’ " —u a with? /m.&}m=f” ﬁwit 7% 5‘7] v6 Questions 11 and 12 are work-out problems (25 pts each). You must Show work to receive credit. 11. Let 9 be a discrete random variable which takes values {0, g, 71', 37"} with equal probability. LethcoseandesinG. ‘U (a) Find the joint probability mass function for X and Y. 9 - if (b) Find the marginal probability mass functions for X and Y. (c) Find the means for X and Y. (d) Find the correlation between X and Y. — i ‘ (e) Are X and Y independent? Are X and Y uncorrelated? (You must justify your answer in terms of your answers to parts (a) — (d)) — ‘HAQ (a\ (XKY) tutu wukves E Cho\l Codi) (—u.o)) Co iii um ﬁauﬂ w» v; t. ' i' ( ' 0 UV C \‘ ‘ " ,.. “M? l:- 1 ?X‘( IV. 133‘) H 3‘ - O .) = t C1\,3&)' L , , L L1v.\$}i= C'h°l ' ‘1 2t (*‘|,_5}) :(O"‘) / ._. 0 Mu QM i’XCM” Z PM (“3‘30 “(‘31)" E i’whl'j’) “35' __ L 3‘; '2 L 7‘1“ F “l i ti -: L :0 : .L 11:0 // 1 L33 // a g :- t l— lxl2-‘ ’- {l «2'3 ‘ _: e\<,2 ': 0 em» Uh EH‘H: 73 73 W13 PHWHO — new Om tn (”WWW 1‘ A” + ot-Utti :5 // Ceh FxTi Mi) :3 01L Li‘(\l;) : ‘7)‘(313 L‘h) __> Mot [MAQPeU—MJ’ 12. Let X and Y be independent Gaussian random variables each with mean 0 and variance 1. Let Z : \/ X 2 + Y2 (this problem arises when, for example, X and Y are the real and complex parts of a certain random phasor, and the amplitude Z is desired) (a) Find E [Z] (Hint: in this part and other parts, you may want to consider switching to polar coordinates and using tables to evaluate integrels). (b) Find Var [Z] . (c) FindPr[Zgl]. QXYL‘I- j) (d) Find fz (2:). 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prevexam2b - EE302 Midterm#2 Section 2 TR 12:00 1:15 Prof...

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