Algorithms for Radio Networks Exercise 12

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Algorithms for Radio NetworksExercise 12
Stefan Rührup sr_at_upb.de
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Exercise 24

• 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
• What is the cumulative probability function PR
  r for the distance R s-t2 between the start
  node and target node if r 1?
• What is the cumulative probability function PR
  rif 1 r v2?
• Compute the corresponding probability density
  function and draw the graph of the function.
• What is the expected value of R?

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Exercise 24
1.) r 1
2.) 1 r v2
t
A2
t
A1
s
s
PR r (8 A1 4 A2) / 4
PR r / ? r2 / 4
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Exercise 24
r
vr2-1
A1 vr2-1
A1
?
t
1
A2
A1
? ?/2 - 2?cos ? 1/r
s
A2
?
A2 ? r2 ?/(2?)
PR r (8 A1 4 A2) / 4
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Exercise 24
F1(r)

• r 1 PR r ? r2 / 4 PR r ? r / 2
  
• 1 r v2 PR r PR r ER
  0.765 Numerical integration of where fR is the
  probability density function (PDF), which is
  piecewise defined by the two functions above

F2(r)
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Exercise 24

• Distribution function (cumulative probability)

PR r
r
F2(r)
1
0
F1(r)
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Exercise 24

• Probability Density Function (PDF) for 0 r v2

PR r
r
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Exercise 25

• Find a counter-example that disprovesfor
  independent random variables X and Y.
• Chose X 1,2 and Y1,2 with PX1 PX2
  1/2 and PY1 PY1 1/2
• EX ? k PXk 3/2EY ? k PYk
  3/2
• EX/Y ? k PX/Yk 9/8

PX/Y X1 X2
Y1 1/4 1/4
Y2 1/4 1/4
X/Y X1 X2
Y1 1 2
Y2 1/2 1
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Exercise 26 (additional exercise)

• 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 EV?
• Show that v EV
• 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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Exercise 26

• PV vmin 1/2 and PV vmax 1/2.
• The average speed is given by
• ED d is fixed. But what is the expected time
  ET?
• If the speed is constant, one needs a time of t
  d/v to cover a fixed distance d.

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Exercise 26

• PV vmin 1/2 and PV 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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Exercise 26

• Show that

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Exercise 26

• What is the minimum of the function x1/x for x gt
  0?(this is a relative minimum, because the
  second derivative is greater than 0)
• x1/x 2 for x1Therefore,

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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 sr_at_upb.de F2.313