Mathematics in Terrains
-
Upload
sofia-fuentes -
Category
Documents
-
view
232 -
download
0
Transcript of Mathematics in Terrains
-
8/11/2019 Mathematics in Terrains
1/32
Some experiments in probability theory - A prelude
to mathematics on terrains
Gopikrishnan C R
School of MathematicsIndian Institute of Science Education and ResearchThiruvananthapuram
September 18, 2014
1 / 15 Gopikrishnan C R Math.On.Terrains
http://find/http://goback/ -
8/11/2019 Mathematics in Terrains
2/32
2 / 15 Gopikrishnan C R Math.On.Terrains
http://find/ -
8/11/2019 Mathematics in Terrains
3/32
Birthday Problem
ProblemWhat is the probability that two persons in a group ofn peoplehave their bithdays on the same date and month?
3 / 15 Gopikrishnan C R Math.On.Terrains
http://find/ -
8/11/2019 Mathematics in Terrains
4/32
Birthday Problem
ProblemWhat is the probability that two persons in a group ofn peoplehave their bithdays on the same date and month?
What is the sample space ()?
= {{x1, x2, . . . , xn}|xi is the birthday of the ith person}|| = 365365 365, n times
= 365n
3 / 15 Gopikrishnan C R Math.On.Terrains
http://find/ -
8/11/2019 Mathematics in Terrains
5/32
Birthday Problem
ProblemWhat is the probability that two persons in a group ofn peoplehave their bithdays on the same date and month?
What is the sample space ()?
= {{x1, x2, . . . , xn}|xi is the birthday of the ith person}|| = 365365 365, n times
= 365n
What is the event space (E)?
E= {{x1,
x2, . . . ,
xn}|xi=xjfor some i,j}
3 / 15 Gopikrishnan C R Math.On.Terrains
http://find/ -
8/11/2019 Mathematics in Terrains
6/32
Birthday Problem
ProblemWhat is the probability that two persons in a group ofn peoplehave their bithdays on the same date and month?
What is the sample space ()?
= {{x1, x2, . . . , xn}|xi is the birthday of the ith person}|| = 365365 365, n times
= 365n
What is the event space (E)?
E= {{x1,
x2, . . . ,
xn}|xi=xjfor some i,j}What we dont require (EC)?EC = {{x1, x2, . . . , xn}|xi=xj for any i,j}
3 / 15 Gopikrishnan C R Math.On.Terrains
http://find/ -
8/11/2019 Mathematics in Terrains
7/32
Probability of same birth dates
|EC| = 365364363 (365n + 1)
= 365!
(365n)!
4 / 15 Gopikrishnan C R Math.On.Terrains
http://find/ -
8/11/2019 Mathematics in Terrains
8/32
f
-
8/11/2019 Mathematics in Terrains
9/32
Probability of same birth dates
|EC| = 365364363 (365n + 1)
= 365!
(365n)!
=
365Pn
|E| = || |EC|
= 365n 365Pn
4 / 15 Gopikrishnan C R Math.On.Terrains
P b bili f bi h d
http://find/http://goback/ -
8/11/2019 Mathematics in Terrains
10/32
Probability of same birth dates
|EC| = 365364363 (365n + 1)
= 365!
(365n)!
=
365Pn
|E| = || |EC|
= 365n 365Pn
Probability ofE=365n 365Pn
365n
= 1365Pn
365n
4 / 15 Gopikrishnan C R Math.On.Terrains
Wh d hi i ll ?
http://find/ -
8/11/2019 Mathematics in Terrains
11/32
What does this convey numerically ?
Final Result
Probability of two persons having the same birth date,p =
1365
Pn365n
Let us do a numerical analysis, how does this function behave.
5 / 15 Gopikrishnan C R Math.On.Terrains
http://c/Users/user/Desktop/Presentation/birthday.nbhttp://c/Users/user/Desktop/Presentation/birthday.nbhttp://find/http://goback/ -
8/11/2019 Mathematics in Terrains
12/32
Number of persons p
1 0
10 0.11694820 0.411438
30 0.706316
40 0.89123250 0.970374
60 0.994123
70 0.99916
80 0.999914
90 0.999994
100 1
6 / 15 Gopikrishnan C R Math.On.Terrains
B ff N dl P bl
http://find/ -
8/11/2019 Mathematics in Terrains
13/32
Buffons Needle Problem
Problem
If a short needle, of length l, isdropped on a paper that is ruledwith equally spaced lines of
distance d l, then theprobability that the needle comesto lie in a position where itcrosses one of the lines is exactly
p= 2ld
Figure : Georges-Louis Leclerc,comte de Buffon
7 / 15 Gopikrishnan C R Math.On.Terrains
http://find/ -
8/11/2019 Mathematics in Terrains
14/32
Figure : Needles over a plane ruled by parallel lines
8 / 15 Gopikrishnan C R Math.On.Terrains
http://find/http://goback/ -
8/11/2019 Mathematics in Terrains
15/32
Let pi is the probability that the needle crosses exactly i timesthe lines.
9 / 15 Gopikrishnan C R Math.On.Terrains
http://find/ -
8/11/2019 Mathematics in Terrains
16/32
Let pi is the probability that the needle crosses exactly i timesthe lines.
Then the expected number of crossings is,
E=p1 + 2p2 + 3p3 + . . .
9 / 15 Gopikrishnan C R Math.On.Terrains
http://find/http://goback/ -
8/11/2019 Mathematics in Terrains
17/32
Let pi is the probability that the needle crosses exactly i timesthe lines.
Then the expected number of crossings is,
E=p1 + 2p2 + 3p3 + . . .
If the needle is short (l d), then p2=p3= = 0
9 / 15 Gopikrishnan C R Math.On.Terrains
http://find/ -
8/11/2019 Mathematics in Terrains
18/32
Let pi is the probability that the needle crosses exactly i timesthe lines.
Then the expected number of crossings is,
E=p1 + 2p2 + 3p3 + . . .
If the needle is short (l d), then p2=p3= = 0
There fore in this case we have,
E=p1
That means we are actually searching for the expected
number of crossings of a short needle.
9 / 15 Gopikrishnan C R Math.On.Terrains
http://find/ -
8/11/2019 Mathematics in Terrains
19/32
Let pi is the probability that the needle crosses exactly i timesthe lines.
Then the expected number of crossings is,
E=p1 + 2p2 + 3p3 + . . .
If the needle is short (l d), then p2=p3= = 0
There fore in this case we have,
E=p1
That means we are actually searching for the expected
number of crossings of a short needle.Let E(l) denotes the number of crossings that will beproduced by dropping a straight needle of length l.
9 / 15 Gopikrishnan C R Math.On.Terrains
http://goforward/http://find/http://goback/ -
8/11/2019 Mathematics in Terrains
20/32
Let l=x+ y, that is the length of the rear part is x andlength of the front part is y. Then by applying linearity ofexpectation
E(l) =
E(x) +
E(y) (1)
10 / 15 Gopikrishnan C R Math.On.Terrains
http://find/ -
8/11/2019 Mathematics in Terrains
21/32
Let l=x+ y, that is the length of the rear part is x andlength of the front part is y. Then by applying linearity ofexpectation
E(
l) =
E(
x) +
E(
y) (1)
Then inductively we shall obtain,
E(nx) =nE(x) (2)
10 / 15 Gopikrishnan C R Math.On.Terrains
http://find/ -
8/11/2019 Mathematics in Terrains
22/32
Let l=x+ y, that is the length of the rear part is x andlength of the front part is y. Then by applying linearity ofexpectation
E(
l) =
E(
x) +
E(
y) (1)
Then inductively we shall obtain,
E(nx) =nE(x) (2)
There fore,E(rx) =rE(x) rQ (3)
10 / 15 Gopikrishnan C R Math.On.Terrains
http://find/ -
8/11/2019 Mathematics in Terrains
23/32
Let l=x+ y, that is the length of the rear part is x andlength of the front part is y. Then by applying linearity ofexpectation
E(l) = E(x) +E(y) (1)
Then inductively we shall obtain,
E(nx) =nE(x) (2)
There fore,E(rx) =rE(x) rQ (3)
Clearly E(x) E(y) ifx y. That is E(x) is a monotonefunction of x.
10 / 15 Gopikrishnan C R Math.On.Terrains
http://find/ -
8/11/2019 Mathematics in Terrains
24/32
Let l=x+ y, that is the length of the rear part is x andlength of the front part is y. Then by applying linearity ofexpectation
E(l) = E(x) +E(y) (1)
Then inductively we shall obtain,
E(nx) =nE(x) (2)
There fore,E(rx) =rE(x) rQ (3)
Clearly E(x) E(y) ifx y. That is E(x) is a monotonefunction of x.
Solving the above equation we shall obtain,
E(x) =cx (4)
Our aim is to find what is this constant c= E(1)
10 / 15 Gopikrishnan C R Math.On.Terrains
http://find/ -
8/11/2019 Mathematics in Terrains
25/32
Consider a polygonal needle of total length l.
Then the linearity of expectation gives E(l) =cl
11 / 15 Gopikrishnan C R Math.On.Terrains
http://find/ -
8/11/2019 Mathematics in Terrains
26/32
Consider a polygonal needle of total length l.
Then the linearity of expectation gives E(l) =cl
Consider a circular needle C of diameter d. Then the total
length of this needle is d. Irrespective of the position bywhich the needle falls, it will make two crossings.
11 / 15 Gopikrishnan C R Math.On.Terrains
http://find/ -
8/11/2019 Mathematics in Terrains
27/32
Approximatethe circular needle from outside by a polygonalneedle Pn and from inside by Pn. Then clearly,
E(Pn) E(C) E(Pn
) (5)
cL(Pn) 2 cL(Pn)
12 / 15 Gopikrishnan C R Math.On.Terrains
http://find/ -
8/11/2019 Mathematics in Terrains
28/32
Approximatethe circular needle from outside by a polygonalneedle Pn and from inside by Pn. Then clearly,
E(Pn) E(C) E(Pn
) (5)
cL(Pn) 2 cL(Pn)
Taking the limit as n tends to infinity, the polygons becomemore and more same as the circular needle. Thus,
c limn
L(Pn) 2 c limn
L(Pn)
cd 2 cd
c= 2
d (6)
12 / 15 Gopikrishnan C R Math.On.Terrains
http://find/ -
8/11/2019 Mathematics in Terrains
29/32
Final Result
Probability that a short needle crosses a line =
2l
d
There fore if we make M trials and N out of them cross a line, then
N
M
2l
d
2l
dN
M
2lM
dN
13 / 15 Gopikrishnan C R Math.On.Terrains
http://c/Users/user/Desktop/Presentation/pi_cal.xlsxhttp://find/ -
8/11/2019 Mathematics in Terrains
30/32
Final Result
Probability that a short needle crosses a line =
2l
d
There fore if we make M trials and N out of them cross a line, then
N
M
2l
d
2l
dN
M
2lM
dN
Let us implement this experiment.
13 / 15 Gopikrishnan C R Math.On.Terrains
Polygonal Approximation
http://c/Users/user/Desktop/Presentation/pi_cal.xlsxhttp://c/Users/user/Desktop/Presentation/pi_cal.xlsxhttp://find/ -
8/11/2019 Mathematics in Terrains
31/32
14 / 15 Gopikrishnan C R Math.On.Terrains
References
http://find/ -
8/11/2019 Mathematics in Terrains
32/32
M. Aigner, K.H. Hofmann, and G.M. Ziegler.Proofs from THE BOOK.Springer, 2010.
L.C Evans.
An introduction to stochastic calculus.W. Feller.AN INTRODUCTION TO PROBABILITY THEORY AND
ITS APPLICATIONS, 2ND ED.
Number v. 2 in Wiley publication in mathematical statistics.Wiley India Pvt. Limited, 2008.
15 / 15 Gopikrishnan C R Math.On.Terrains
http://find/