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EE 416 Homework Assignment 1 Revised
1. (20 points)
(a) Let A, B be events. Draw a Venn diagram that illustrates P (A ∪ B) =
P (A) + P (B) − P (A ∩ B)? How does this equation simplify when (i) A and
B are disjoint or (ii) when A and B are independent. For arbitrary A, B, C,
extend this result to P (A ∪ B ∪ C)
(b) If P (A) = a, P (B) = b > 0, then P (A | B) ≥ (a + b − 1)/b. Prove.
(c) If P (A) = 1/3, P (B) = 1/4, can it be true that A ⊂ B? Prove.
(d) For any N = 1, 2, 3, . . . , justify the “Union Bound,”
P (A1 ∪ A2 ∪ . . . AN ) ≤
N
X
P (Ak ).
k=1
First, work this out for N = 2 and N = 3. Then use induction on N .
When will this bound be a good approximation? When will it be a useless
approximation?
2.
(20 points) (a) This is a problem in geometric probability, and
you need the Table.A dime, penny, etc have the given diameters. Play a
game, where we toss each type of coin, each in turn, at random onto a huge
infinitely large square grid. This grid has lines of width W = 5mm and
each are separated by a distance D, as measured from inside-to-inside of the
lines.
Coin
Roosevelt Dime
Lincoln Cent
Jefferson Nickel
Washington Quarter
Kennedy Half Dollar
Eisenhower Dollar
Diameter
17.9 mm
19.0 mm
21.2 mm
24.3 mm
30.6 mm
38.1 mm
Sketch a picture of the grid and label. You toss the coin and win if you
do not touch a grid line.If we want the odds to be 50; 50, what should the
Distance D be, given the diameters and widths are specified as above. Get
a general formula, and evaluate for each coin in the list.
3.
(10 points) (b)Geometric Probability. It is equally likely that two
signals arrive at a receiver at any time 0 < t < T , where T is known. Assume
that the receiver is jammed if both signals arrive within time difference τ .
Sketch the sample space and by computing areas on a T ×T square, (i) find a
general formula for the probability of jamming, PJ , and (ii) the probability
that both signals are received clearly, PC . Show (iii) that for very small
γ := τ /T, PJ ≈ 2γ. As T → ∞, PJ → 0. Explain (iv) why this is true.
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4.
(10 points) A communications link is given, connecting the source,
S, to the receiver, R. It consist of N identical series elements a, followed
by M parallel elements b, as depicted. Each element a, b fails independently
of the others. Let Ak be the event that ak is working, with probability
P (Ak ) = pk , and similarly for Bk , with P (Bk ) = qk . Using unions and
intersections of appropriate events, (i) find a general expression for the event
O, that the link is out (outage). Let A denote the event that the series link
is working and B that the parallel link is working. Then (ii) set pk = 0.99,
qk = 0.50, and find and set M so that P (B) = 0.99. and find PO , the
corresponding probability of outage, for 1 ≤ N ≤ 10, (iii) how large must
N be to assure that PO ≤ 0.1 ? (iv) What happens if N > 10 ? Solve with
MATLAB.
5. (40 points) We discussed the Birthday Problem in class. Birthdays
are randomly distributed over the year, D = 365 days. How many people P
must you query so that the probability of two or more birthdays matching
exceeds 0.5.
Write a Monte Carlo Simulation of this problem, by looping over T =
1000 independent trials. Show that your result agrees with the theory. Extend your simulation to estimate the probability that two or more birthdays
either match or are adjacent.
This is a common problem in uncoordinated access to a shared medium.
The people represent nodes trying to access slots (the days). If two or
more nodes try and access the same slot, a collision occurs and they must
retry. The question is the maximum density P/D that allows uncoordinated
access. To draw P people’s birthdays randomly on N days use
b = 1 +floor(rand(P,1)*N);
The flow is
knt = 0;
for t = 1:T
Generate the Brithdays of p people in [1:N]
See if any birthdays satisfy the conditions,
if so set
knt = knt+1;
end
pest = knt/T;
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