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The Fundamental Theorem of Probabilistic Number Theory - Shared screen with speaker view
Jake Robb
12:53
I can
Derek Liu
12:54
yes
Gavin Abbe
12:56
yea
Andrew Park
12:58
yes
Justin Kang
12:58
yes
Nan Feng
12:58
yes
Jewoo Suh
12:59
yes
Sebastian Weinberger
13:00
Yes
Eddy Li
13:01
yes
ChangYu
13:01
yes
Sangyeop Song
13:03
yep
Sheldon Tan
13:10
yes
espen slettnes
14:05
Handout: https://espen.slett.net/BMC_lectures/Probabilistic_Number_Theory/2020-06-15.pdf
Derek Liu
16:12
half of the girls
Andrew Sylvester
16:13
g/2
ChangYu
16:16
g/2
Derek Liu
16:40
b/2
Derek Liu
16:54
because it all depends on whether one specific fgirl gets in
Derek Liu
16:57
like fix the rest
Derek Liu
17:01
then like 1/2 chance of that girl being in
Derek Liu
17:52
linearity
Derek Liu
18:35
because the average is half the students
Eddy Li
20:15
not bipartite
Derek Liu
20:16
no because there is a triangle
Sheldon Tan
20:16
no
Jake Robb
20:19
No. There is a triangle
Sebastian Weinberger
20:29
No; it has an odd cycle between 1, 3, 5
Derek Liu
20:33
bipartite triangles only have cucles of even size
Derek Liu
20:40
there is no "labeling" of a triangle
Sheldon Tan
20:48
because triangle 135 for example whatever 1 is 3 and 5 must be the opposite but then 3 = 5
Derek Liu
21:41
yes
Eddy Li
21:41
yes
Andrew Sylvester
21:42
yes
Sebastian Weinberger
21:43
Yes
Leo Xu
21:43
yes
Jake Robb
21:45
yes
ChangYu
21:46
yes
Derek Liu
22:47
shouldn't the question say "at least m/2" instead of "at most m/2" then?
Sheldon Tan
23:01
yes
Sheldon Tan
23:03
it does
Eddy Li
23:06
1,2
Derek Liu
23:13
15, 45, 34
Andrew Sylvester
23:17
1-5, 4-5, 3-4
Vinci Ho
23:19
wait, is a subgraph part of a graph or the edges turning into vertices?
Derek Liu
23:57
2 is opposite gender of rest
Eddy Li
24:01
2=B 1=2=4=5=g
Eddy Li
24:10
I mean 1=3=4=5
Nathan Barnes
25:13
1/2
Derek Liu
25:15
each edge has 1/2 chance
Derek Liu
25:23
because bb and gg each have 1/4 chance
Derek Liu
25:28
so together it is a 1/2 chance of removed
Derek Liu
26:03
in theory, could you get a stronger bound than m/2 by consider colorings with equal b and g vertices? (or 1 off)
Sebastian Weinberger
28:56
Yes but barely I think… we could probably get (m+1)/2
Nathan Barnes
29:09
27
Derek Liu
29:14
just over 1/4*1/4*432=27
Leo Xu
29:15
108*1/4
Derek Liu
29:18
because we don't consider identity
Nathan Barnes
29:22
there are 4 colors
Derek Liu
29:22
so we get more than 27 ithink
Derek Liu
30:22
but we can get 28
Derek Liu
30:25
because identity has 0
Derek Liu
30:59
the average of the other 431 is just over 27
Nathan Barnes
31:51
7 but we can get 8
Derek Liu
31:55
^
Eddy Li
32:31
8/2+1=3=triangles
Derek Liu
33:17
but this seems like a very strict bound
Derek Liu
33:33
actually 428 should work but requires more care
Derek Liu
33:43
actually it wont
Derek Liu
36:50
standard dev
Eddy Li
36:53
st. deviation?
Derek Liu
38:58
P(y at least b) is at most E(Y)/b?
Nathan Barnes
39:40
y≥b <-> |x-E[x]|≥a
Derek Liu
40:06
*at least a
Derek Liu
40:08
?
Eddy Li
40:16
b P(y>=b) is at most Var (x)
Derek Liu
40:20
variation?
Derek Liu
43:06
well we have the lambda^2 in the denominator already
Derek Liu
43:10
and the deviation of lambda
Derek Liu
45:17
because
Derek Liu
45:24
S_A is the sum of the corresponding x_i
Nathan Barnes
45:29
we get heads on the x_i in A and tails on the ones not in it
Derek Liu
45:37
so its essentially S_A is the sum of a subset of {x_i}
Nathan Barnes
47:59
0
Eddy Li
48:20
why is it zero
Derek Liu
48:34
^ its 0 because the x_i sum to 0
Eddy Li
48:41
thx
Sebastian Weinberger
49:16
The first is 1/2
Derek Liu
49:36
each x_i has 1/2 chance of being in
Sebastian Weinberger
50:12
Should be 0?
Sebastian Weinberger
50:16
Not sure...
Derek Liu
50:28
like -1/something?
Eddy Li
51:41
1/4
Derek Liu
51:47
E(x_i)E(x_j)
Eddy Li
52:27
(x_i)/2
Eddy Li
52:28
?
Derek Liu
53:29
sum of x_ix_j is -1/2 I think?
Derek Liu
54:06
1/2((x_1+...+x_n)^2-x_1^2-x_2^2-...)
Derek Liu
56:31
so |S_A| at least lambda with probability at most 1/4lambda^2
Derek Liu
56:44
so S_A at least lambda with probability at most 1/8lambda^2
Eddy Li
57:22
what does the "| |" sign mean
Derek Liu
57:26
abs value
Eddy Li
57:36
is it abs. val?
Derek Liu
57:56
1/(4lambda^2)
Derek Liu
58:14
but since we are asked for S_A at least lambda we need to halve that
Derek Liu
58:31
and we know at least lambda and at most -lambda are equally likely because complement sets
Derek Liu
59:08
2^n
Derek Liu
59:54
a set and its complement have negative sums
Derek Liu
01:00:07
so each S_A at least lambda can be paired with an S_B at most -lambda
Derek Liu
01:00:10
so equally likely
Derek Liu
01:00:48
but when does equality hold for markov
Derek Liu
01:00:52
because problem asks for equality
Eddy Li
01:01:06
is the hw #7, #9, #5
Nathan Barnes
01:01:35
How do we know that this doesn't just work on average?
Derek Liu
01:01:47
this is on average
Derek Liu
01:01:52
over all subsets though
Derek Liu
01:02:16
we're proving that at most 1/(8lambda^2) of subsets work by EV
Nathan Barnes
01:02:23
what if more subsets have a higher sum because of random chance?
Derek Liu
01:02:52
it cant
Derek Liu
01:02:55
what we found was
Derek Liu
01:03:02
a strict probability of 1/(8lambda^2)
SheldonTan
01:03:27
Thanks!
Sebastian Weinberger
01:03:27
Thank you!
Derek Liu
01:03:28
Thank you!
Jake Robb
01:03:32
Thanks!
Eddy Li
01:03:35
Thank you!
Sangyeop Song
01:03:35
Thank you
ChangYu
01:03:36
Thank you
Zihongbo Wang
01:03:37
Thanks
Leo Xu
01:03:38
Thank you
Owen Xu
01:03:39
Thank you!