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spring10-mt1-soln

# spring10-mt1-soln - COMER February 8 2010 ECE 302 Exam 1 1...

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Unformatted text preview: COMER February 8, 2010 ECE 302 Exam 1 1. Enter your name and signature in the space provided below. Your signature certiﬁes that you will not engage in any act of academic dishonesty during the taking of this exam. 2. You may not use a calculator or any other reference materials. This exam consists of four problems. For the ﬁrst three problems, no partial credit will be given. For the fourth problem, partial credit will be given at the discretion of the instructor. (4) Name: 3 O U “n Mg Signature: No Partial Credit You must put your ﬁnal answers in the boxes provided. 1. (20 points) Four marbles, numbered 1,2,3, and 4, are placed in a box. One of the marbles is drawn randomly from the box and its number, N1, is noted (so N1 = 1, 2, 3, or 4). An integer N2 is then selected at random from the values 1, . . . ,Nl. The outcome of this experiment is the ordered pair (N1 7 N 2)‘ (a) (5 points) Write the sample space for this experiment. Fiswéi: i) D 914/ iii/€90 f, [ 27% m i, r f‘ "‘ _ 42% 2W 2 it (b) (5 P0ir{t8)Wfi t 6 even “Ii/Iafb' e2 is selected”. j ~ (d) (5 points) Write the event “Marble 2 is selected and N2 = 3”. J No Partial Credit You must put your ﬁnal answer in the box provided. 2. (30 points) Let E, F, and G be three events in an event space, with probabilities P(E) = 103, P(F) 2: pp, and 2 pg. (3) (5 points) Find an expression for the event that only E occurs. (b) (5 points) Find an expression for the event that at most one of the three events occurs. "EEK-liver 5M“? W amt, <2 yea/mt (cf Li 01f ‘Answezéz €121 {Niféi A (e) (5 points) Find P(E LJ F) if E and F are independent. e. Wm mm w P/ A l7é22f>+W33 men ()2: «:3 (t) (5 points) Find P( U F U G if E,F, an G ar pairwise mutually exclusive. ibwnmﬁ :3 No Partial Credit You must put your ﬁnal answer in the box provided. 3. (15 points) A simpliﬁed model for the movement of the price of a stock supposes that on each day the stock’s price either moves up one unit with probability p or it moves down one unit with probability 1 — p. The changes on different days are assumed to be independent. (a) (5 points) What is the probability that after twoﬂdays the stock will be at its oiji inal price? , Ea] ““}"VU§ dam? gr ﬂaw if? 1 W (b) (5 points) What ‘is the probability t at aﬁerﬂ three days the stock’s price will have increased by one unit? a ‘3“ m. «:23? ‘ "7:; (“x a. c l ,. i I é"%"~~*7[riw W N up) clamp"; 2 iii-7‘5? U j: at?) upjgijgﬁ/Uyr Witt/w} up} \ _ (c) (5 points) Given that fter three da 3’ the stock’s price has increased by one unit, what is the probability Eat it went up on the ﬁrst day? ' l} . i) cm W 2 Ci, ---- «A i i f) .......................................................... " ' _ a“) (a, ‘ I , ’ a , h t 1:: l (a utpj 04km 2»; Muff“ U { up, My Ala-«mm i— i Partial Credit Problem You must completely justify your solution to get full credit. Partial credit will be given at the discretion of the instructor. 4. (35 points) Let A and B be events in an event space. (a) (10 points) If A and B are independent, are they mutually exclusive? Explain. (b) (10 points) Show that if P(A|B) > P(A), then P(B[A) > P(B). (c) (15 points) Show that ifP(A|B) = 1, then P(BC[AC) = 1. {1(1) N5 {Vt (EM/'th if“ g 1‘! 1" f “ i, w ~¢mwMe ﬁﬂdﬂﬂﬁ @«1‘31 “MN m" 7 .‘ g. x g ﬂ o i. it “H”??? a” ,ﬂ it A H n 1% 5 <2; r/W> a... _/ i eel/’lxi,fi”éi...(i,;0i\jg» {3K {VQ ...
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spring10-mt1-soln - COMER February 8 2010 ECE 302 Exam 1 1...

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