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Unformatted text preview: THE UNIVERSITY OF TEXAS AT AUSTIN
Dept. of Electrical and Computer Engineering Midterm #1
Date: October 5, 2010 Course: EE 351K
Name: Last, First 0 The exam will last 75 minutes. a Closed book and notes. You may consult only one double sided page of handwritten
notes. 0 No calculators are allowed. 0 All work should be performed on the exam itself. The backs of the pages may be used
as scrap paper. To avoid confusion, please highlight your ﬁnal solution in the space
provided. 0 Fully justify your answers for more rational grading. 0 Good luck! Problem 1. [10 points] We play the following game: first, we toss a coin with P(tails) = q.
If the coin comes up heads, we roll a 4sided die; otherwise, we roll a 6sided die. Let X be a
random variable indicating the outcome of the coin toss: X = 0 if the toss is heads, X 2 1 if
the toss is tails. Moreover, let Y denote the random variable corresponding to the outcome
of the roll of a die (note: Y takes values from the set Sy = {1, 2, 3, 4, 5, 6}). (a) (3 points) Determine the joint probability mass function p Xy. (r i “Pm mm = § at m, ye rims}
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Z) 7" Problem 2. [10 points] A factory manufactures computer chips. Each chip may turn out
defective with probability (1, independent of any other chips. (a) (2 points) What is the probability that a shipment of n chips will contain It defective
chips? 3L3 3;)“ €35“ o? o. MMQQQX‘ mil/V: ( ,P (t: “ii (3% Mk 2 AIM {A MOXXMAQ (b) (3 points) Suppose that chips are tested in sequence, one by one, until a total of k
defective chips are discovered. Find the probability that the kth defective Chip is found
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P(test says chip OK I chip defective) = P(test says chip defective  chip OK) = t, , for some 0 <2 t < 1. Suppose that each chip is tested 10 times independently, and
rejected if at least 6 0f the tests come up defective. Given that a chip is rejected, ﬁnd the probability that it is defective. A: iv (5 6V (wee (Loewe w? Aggﬁgklfwe We, sway/dc {My}; {Me Sta? L5 guea‘limw: ?(bl k ‘1 z. Wm M WNW mm? (Ew‘lﬁs mm)
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nodes A and B, connected with a single link. At any given time, the link is up with probability :— 02. Starting from time t = 0, node A attempts to transmit a data packet to node B.
The transmission attempts are exactly 1 second apart (1.6., the ﬁrst transmission attempt is
at precisely t = 1 second, the next one is at t = 2 seconds, etc.). Let Y denote the time that node B waits for the arrival of the packet. [Note: if the link is up when node A attempts to
send the packet, the packet is transmitted instantaneously] (a) (3 points) What is the expected time node B waits for the arrival of the packet? E {:2 (3% gthCBS (b) (3 points) To improve the reliability of the network, another link is added in parallel
to the ﬁrst one. At any given time, this new link is up with probability 0 < q < 1.
The packet can be transmitted if either one of the links is up. We are told that the expected time node B waits for the arrival of the packet is E[Y] = 169 seconds. Find q. i \M ?C\TV¢M WM 2 A c 0"?) U7»)
W 291*G’gCAHZ): (c) (4 points) Consider again the single link scenario. We are told that node B waits at
least 4 seconds for the packet arrival (i.e., Y _>_ 4). Given this information, what is the expected time node B waits for the packet (i.e., find E[Y1Y 2 4])? \LD Q CEvaxk, E‘Ck‘tcwk‘b
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