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Unformatted text preview: EE 131A
Probability
Instructor: Lara Dolecek Fall 2010 Final
Friday, December 10, 2010 Maximum score is 200 points. You have 180 minutes to complete the exam. Please Show your work.
Good luck! Your Name: Your ID Number: Name of person on your left: Name of person on your right: Problem Score Possible l
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f“ Total 200 I 1. (15 pts) You select two numbers (without replacement) from the set {1,2,3,4, 5}.
What is the probability that the sum is at most 6 given that the smaller number is at
most 3 7 Somok. ‘ M6101 vamzh (1,2), <(,%>, (MU (LE), (23)
(NH w?) 13¢). WA”) :35) [[42le alga)? my“? )9) :7 L 2. (10+5 pts) Suppose X is Gaussian with mean m and variance 02. (a) Find E[(X h m)3].
(b) Find pdf of Y 2 8X . Random variable Y is called log—normal. Amie”, , 3. (lO+5+10+10 pts) Suppose X is Poisson r.v. with parameter A. That is, pinf of X is
given by P(X : k) : ”\k—feiﬁxfor k=O,1,2,3,... : (a) Calculate the characteristic function (13X (w) of X. You may ﬁnd useful the fol—
lowing: et : 2:0 7%. b Calculate the 1 r n and var' nce of X.
< > s? is
(c) Verity the inequality (d) Suppose that Y and Z are also Poisson r.v.s with parameter A, and that X , Y, Z
are independent. What is the distribution of W 2 X + Y + Z ? 3;. 4. (10 pts) True or False. Circling the correct answer is worth +2 points, circling the incorrect answer
is worth w1 points. Not circling either is worth 0 points. (a) If X and Y are uncorrelated and exponential then X and Y are independent.
TRUE {FALSEQ
(b) If X is X N N(0,1) and Y 2 X2 then Y is also Gaussian.
TRUE {EALsE5
(0) Suppose X and Y are independent. Then E[X2Y3] : E[X2]E[Y3].
{TRUE;E FAL$E
(d) Suppose X N N(4, 4) and Y m N(27 2) are X and Y are independent. Let
Z 2 X — Y. Then Z N N(2,2).
TRUE fﬁALgi (e) If X and Y are independent and Y and Z are independent7 then X and Z are
independent. ‘ TRUE iTALSE! Ur 5. (10+10+5 pts) Consider X and Y be jointly distributed according to the joint density
f(.r, y) = My ~ $)27 for 0 g a: g y g 1 and zero elsewhere7 and Where It is a constant. (a) Find the constant k: to make fX,y($, y) a valid joint pdf.
(10) Find fX(:v).
(C) Find .fXjY($ly)~ _ !\_
; a5 \ "‘7‘“ 6. (5+5+5+5+5 pts) Suppose X and Y are jointly Gaussian with E[X] = 4 and E[Y] = 2
and with covariance matrix 2 1
Kw —~ [ 1 4 J
. a b 1 1 d —b
You may find useful the fOllOWID‘Ti if A then A = _
C d Id be _( a (a) (b) Suppose Z — 2X + Y. What is fz(z ). (0) Suppose also W: X— Y. W hat is fZW(z w) 7
(d) What is fXZ(X Z)? (6) What is pXZ? \{wt AU Z. r . , gr» 7. (15 pts) Let X1, X2, X3.” be independent and uniformly distributed on [071]. Calculate . v X X " X77.
11mm~>oo<bm( 1)+51n( ng)+ +bln( )> 8. (10+10 pts) Suppose you have two decks of n cards, numbered 1 through n. The two
decks are shufﬂed and the cards are matched against each other. We say that a match
occurs at position i if the ith card from each deck has the same number. Let Sn denote
the number of matches. (a) Compute E(Sn).
(b) Compute VAR(S,,,). 9. (20 pts) Let X1, X2, X3... be independent and uniformly distributed on {0,1}, and let
a be a constant. Express the following limit —  (1) ”$00 7”], lim P (/1; X1+... +Xn 1‘ >
W >a in terms of Q(.L') function (Recall that 62(33) = P(X > at) for X standard Gaussian). 10 10. (5+5+10 pts) A communications channel is used to transmit bits of information. Of
the transmitted bits, 60% are ’1’ and 40% are ’0’. When the transmitter sends a ’1’, the input voltage is v = 5, and when a ’0’ is sent the input voltage is v 2 W5. The
noise N depends on the input voltage: it is uniformly distributed on [$38, 8] when U = 5
(Le? when a ’1’ is sent), and it is uniformly distributed on [~v5, 5] when U : ~~5 (i.eW when a ’0’ is sent). The decision threshold of the receiver is at a certain voltage T. This means that the
receiver at the other end decides that a ‘1’ was sent if U + N > T, and that a ’0’ was
sent if v +~ N S T. (a) Find the probability that the receiver decides a ’1’ was sent, given that the trans—
mitter sent a ’0’ for the case where T x 0. (b) Find the total probability of the receiver error for T 2 0. (c) Find the value of the decision threshold T which minimizes the total probability
of receiver error. 11‘ ...
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