131A_1_Final Solution

# 131A_1_Final Solution - Problem 1(20 pts The cumulative...

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Unformatted text preview: Problem 1. (20 pts) The cumulative distribution function (cdf) of the random variable X is given by 0 ifx<0 30/2 ifOSaz<1 FX(x) = 2/3 if1 S11: <2 11/12 if2£x<3 1 " if3<x ._\ _ V) (a) (4 pts) Plot the cdf FX (f I) \ Compute: (b) (4pts) P X < 3]. (C) (4pm) P X =1]. (d) (4pl‘s) P X > 1/2]. (c) (4pts)P 2 < X S 4]. r—ar—ﬁr—qﬁ —‘_\ V \i ._._./ f "3 Problem 2. (15 pts) (3) (5 pm.) A space craft has 20,000 components. The probability of any one component being defective is 1041, The m1331on Will be 1n danger if ﬁve or more components become defective. Find the probability of such an event. Use Poisson approximation. (b) (10 pts) Ms. Jones ﬁgures that the total number of thousands of miles that an auto can be driven before it would need to be junked is an exponential random variable with parameter /\ = ——, i.e., the average lifetime mileage of an auto is 20 thousand miles. Mr. Smith has a used car that he claims has been driven only 10 thousand miles. If Ms. Jones purchases the car, what is the probability that she would get at least 20 thousand additional miles out of it? Repeat under the assumption that the lifetime mileage of the car is not exponentially distributed but rather is (in thousands of miles) uniformly distributed over [0, 40]. .-_ 0 “x C) / ‘ m. \ ‘ . “fa f I; , 1 x I (5?» vx~ VJ“? FF? 3" \‘“'/ ,L» r A (71: U 1‘ \{ A0 ( :: 2’, I ‘5 W7 t / I I i d) AK \"” ‘\ w “1/ C M 'A a " i t 'T l 2 r. M txt": m. Mélm a: ’l/jj 5 l}; ‘v : \ZZO F. ,k (I .32, \ I» (5 Cf” ta? .7 r, U P 1 m t ‘l i \ . L ta ta: ﬂ \ 9 g i [K abs en We”) gaudy * L 1 (,7. WW , w W - . i \V I }:V\W 7A/i/wi i J ff ./\ vi \V/L kmva l) r' n, X ~ F’ W “ x q a“ [v H11“; PLiy‘7VKC/‘s Mi 4 1 ‘7; 7 A, A/\ [/\ > i I {Z ‘ e \. , I; , A I X [f .‘ I! A“) i i, {\N If; 5" L j—M‘N ‘ “1‘ - Aw ~ ' ‘ , Wu“. TN ‘w I“ K \4.\ lo) / ‘\‘ AS) <‘ l \\/ {Vim/J i ii I \ L\*__L__ “(I/WM, L7 \$5773 ‘ X719 E" “M” A? E I! I l J iﬁ’ﬁ L_1 l .\i / \$E/‘i‘ JJ 3!} r/ - "I if”? 1M“? ’TT/M. it 379/;— if" t7 7%; I . 4 E/ﬂ / f) ,J \ 9 f , » D - r» . D J; ‘7’ ; ~ I L ' a. -’ \ Ix.» __ vitalitypts ‘J. J)“ i ( :4. gimx) _ ﬂ ( K. , ,t Problem 3. (15 pts) (3) (5 pts) The random variable X is uniformly distributed in the interval [0, 1]. Find the density function fy(y), whereY : —lnX. (b) (10 pls) We place at random and independently N points in the interval [0, 100]. The distance from 0 to the ﬁrst random point is a random variable Z. Find Fz(z). Hint: Recall that P[Z g z] = 1 — P [Z > z]. Also the probability that any of the points is 33 away from 0 is — 1— Where should each random point be located so that Z > z? 4 100' h L I U , x \ é : P bl " “ v3 fix. L “a i r“? z a m > a E r \ w e” \ I , a .. a; I, I I A i \f’l II it) E L/ i 0 fl, s...“ ‘t/ L, i it, } , ' / \Wiiﬂ/ té/i rm, ’3 in» 5 q 6 a V ,, M’ I j)“ Xg/Eéw; m—.\_\ r 5/ \j i” /7 , A ﬂ " l ,ng Problem 4. (20 pm) (a) (8 pts) To determine the effectiveness of a certain diet in reducing the amount of cholesterol in the bloodstream, 100 people are put on the diet. After they have been on the diet for a sufﬁcient length of time, their cholesterol count will be taken. The nutritionist running this experiment has decided to endorse the diet if at least 65 percent of the people have a lower cholesterol count after going on the diet. What is the probability that the nutritionist endorses the new diet, if in fact, it has no effect on the cholesterol? When a diet has no effect on cholesterol, then assume that each person’s cholesterol will go down randomly, that is with probability 1/2. Use the Central Limit Theorem, and @(30) = 0.99865. (b) (12 pts) Two types of coins are produced at a factory: at fair coin and a biased coin that comes up heads 55 percent of the times. We have one of these coins but do not know whether it is a fair coin or a biased one. In order to ascertain which type of coin we have, we shall perform the following statistical test: We shall toss the coin 1000 times. If the coin lands on heads 525 or more times, then we shall conclude that it is the biased coin, whereas if it lands heads less than 525 times, then we shall conclude that it is the fair coin. (i) If the coin is actually fair, what is the probability that we shall reach a false conclusion? (ii) If the coin is actually biased, what is the probability that we shall reach a false conclusion? :1 Use the Central Limit Theorem. 620/25) 2 0.057 and Q(1.6) m 0.054. Problem 6. (15pfs) A communication channel accepts an arbitrary voltage input 2) and outputs a voltage Y = ’U + N, where N is a Gaussian random variable with mean 0 and variance 02 = 1. Suppose that the channel is used to transmit binary information as follows: to transmit 0 input —1 to transmit 1 input +1 The receiver decides a O was sent if the voltage is negative and a 1 otherwise. (a) (5 pts) Find the probability of the receiver making an error if a O was sent. (Express your answer in terms of Q function ) i (b) (10 pts) Now assume that in order to have a more reliable system each bit is transmitted 5 times repeatedly. The receiver uses majority decoding i.e. in each transmission the receivers makes decision about an individual bit according to the above rule. If the majority of the decisions is 1 the receiver decides that a 1 was sent, otherwise the receiver decides that a zero was sent. Suppose the input bit was a O and transmitted ﬁve times. What is the probability of receiver making an error ? What independence assumption are you making ? WWW y w a Problem 6. (15pm) Consider two independent random variables X and Y, where X is Gaussian with mean m X z 0 and variance 0% = 1, and Y is Bernoulli with p = 0.5. Now deﬁne the random variable Z as X ifY=1 Z: ' —X 1fY:O. (a) (10 pts) Find the cdf of Z (Hint: Find PiZ g 2] using the Total Probability Theorem, and express it in terms of <I>(-) and/Or 620)). What kind of random variable (e.g., exponential, Binomial, Gaussian, Poisson) is Z? Calculate the mean and variance of Z. (b) (5 pts) Are Y and Z independent? (Hint: Find F Z(z|Y = 1) and F 2(z1Y = 0) and compare them with 172(2).) t . i. ii” m i .. it“;:24;\i’ﬂmni{Hm/W} Wt i i «:2; e L “ ~ ~ Kg; i .2”, ‘ V “(’11 I L//€./ V 6 V” f k s i My“; Tigtﬁig j l ‘ a , "‘"i I n l / LYiUrQa’W: - 7 anﬂﬁ L j A / in. g{ ' y \ Q ENxWOH} / s / [I a L Capri Haj O] ‘vgéw’f’i {\1 \\ l/7 " ‘ \‘ p tr i r \i 1 l a / f 27 2 x / ” ., h H (v n m new; ~ ...
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131A_1_Final Solution - Problem 1(20 pts The cumulative...

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