HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity
Algorithms for Radio Networks
Exercise 12
Stefan Rührup
[email protected]
HEINZ NIXDORF INSTITUTE
Exercise 24
University of Paderborn
Algorithms and Complexity
• Assume that a start node s in the center (0, 0) of a square
[−1, 1] × [−1, 1] of edge length 2 chooses uniformly at
random a target node t in this square
1. What is the cumulative probability function P[R ≤ r] for
the distance R = |s-t|2 between the start node and target
node if r ≤ 1?
2. What is the cumulative probability function P[R ≤ r]
if 1 ≤ r ≤ √2?
3. Compute the corresponding probability density function
and draw the graph of the function.
4. What is the expected value of R?
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HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity
Exercise 24
1.) r ≤ 1
2.) 1 ≤ r ≤ √2
t
t
A2
A1
s
P[R ≤ r] =
/
=  r2 / 4
s
P[R ≤ r] = (8 A1 + 4 A2) / 4
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HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity
Exercise 24
√r2-1
t
A2
r

A1
1
A1
s
A1 = √r2-1
 = /2 - 2
cos  = 1/r
A2

A2 =  r2 · /(2)
P[R ≤ r] = (8 A1 + 4 A2) / 4
=
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HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity
Exercise 24
• r ≤ 1: P[R ≤ r] =  r2 / 4
=: F1(r)
P[R = r] =  r / 2
• 1 ≤ r ≤ √2: P[R ≤ r] =
=: F2(r)
P[R = r] =
E[R] ≈ 0.765
Numerical integration of
where fR is the probability density function (PDF),
which is piecewise defined by the two functions above
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HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity
Exercise 24
• Distribution function (cumulative probability)
P[R ≤ r]
r
0
F1(r)
F2(r)
1
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HEINZ NIXDORF INSTITUTE
Exercise 24
University of Paderborn
Algorithms and Complexity
• Probability Density Function (PDF) for 0 ≤ r ≤ √2
P[R = r]
r
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HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity
Exercise 25
• Find a counter-example that disproves
for independent random variables X and Y.
• Chose X = {1,2} and Y={1,2} with
P[X=1] = P[X=2] = 1/2 and P[Y=1] = P[Y=1] = 1/2
• E[X] =  k · P[X=k] = 3/2
E[Y] =  k · P[Y=k] = 3/2
• E[X/Y] =  k · P[X/Y=k] = 9/8
X/Y X=1 X=2
P[X/Y] X=1
X=2
Y=1 1
2
Y=1
1/4
1/4
Y=2 1/2
1
Y=2
1/4
1/4
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HEINZ NIXDORF INSTITUTE
Exercise 26 (additional exercise)
University of Paderborn
Algorithms and Complexity
• An object moves with a constant speed for a fixed distance
d. The speed V is chosen uniformly at random between
either vmin or vmax, i.e. the speed is vmin with probability 1/2
and vmax with probability 1/2.
– What is the average speed v?
– What is the expected speed E[V]?
– Show that v ≤ E[V]
If the speed is constant, one needs a time of t = d/v to cover
a fixed distance d.
The average speed is given by
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HEINZ NIXDORF INSTITUTE
Exercise 26
University of Paderborn
Algorithms and Complexity
• P[V = vmin] = 1/2 and P[V = vmax] = 1/2.
• The average speed is given by
E[D] = d is fixed. But what is the expected time E[T]?
If the speed is constant, one needs a time of t = d/v to cover
a fixed distance d.
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HEINZ NIXDORF INSTITUTE
Exercise 26
University of Paderborn
Algorithms and Complexity
• P[V = vmin] = 1/2 and P[V = vmax] = 1/2.
• The expected speed is
The expected speed is also given by
So, this is another example, where
(see Exercise 25)
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HEINZ NIXDORF INSTITUTE
Exercise 26
University of Paderborn
Algorithms and Complexity
• Show that
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HEINZ NIXDORF INSTITUTE
Exercise 26
University of Paderborn
Algorithms and Complexity
• What is the minimum of the function x+1/x for x > 0?
(this is a relative minimum,
because the second derivative
is greater than 0)
• x+1/x = 2 for x=1
Therefore,
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HEINZ NIXDORF INSTITUTE
University of Paderborn
Algorithms and Complexity
Thanks for your attention!
End of the lecture
Mini-Exam No. 4 on Monday 13
Feb 2006, 2pm, FU.511 (Mozart)
Good Luck!
Stefan Rührup
[email protected]
F2.313