1
00:00:03,725 --> 00:00:06,407
So, we have two equations...ok?
2
00:00:06,407 --> 00:00:09,896
One equation tells us the relationship
3
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between the two Lagrange multipliers
4
00:00:11,750 --> 00:00:12,973
In particular, it tells us "Z"
5
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is a function of Lambda 1
6
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Ok?
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And the second equation here
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two equations, two unknowns
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will allow us to solve for Lambda 1 itself
10
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So, "Z" here (unintelligible)
11
00:00:26,872 --> 00:00:28,445
But if you remember how your geometric
12
00:00:28,445 --> 00:00:30,101
series work, you can actually write this
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out because this, the first couple terms
14
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just to drive your intuition, look like
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one plus 'e' to the negative lambda one
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plus 'e' to the negative 2 lambda one
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and so forth.
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And so we know the geometric series
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of that form by a very nice argument
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that we won't do here.
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Sum to one over 1 minus 'e' to the
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negative lambda 1.
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So we know already how to reduce this
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infinite sum to a very elegant form.
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The next thing I'll point out, because I'm
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an expert at this...I'm always, I'm always
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gonna...going to, uh,
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eventually (unintelligible) from this
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We like to call 'z' the Partition Function
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Ok? 'z' here is the sum over these terms
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00:01:05,941 --> 00:01:08,542
here. And so one thing you notice about
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00:01:08,542 --> 00:01:10,251
'z' about the partition function is if you
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take the derivative of 'z' with respect
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00:01:12,365 --> 00:01:14,861
'i', look what happens. The derivative
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00:01:14,861 --> 00:01:18,550
of 'z' - sorry, we just got to lambda 1 -
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00:01:18,550 --> 00:01:19,816
derivative of 'z' with respect to lambda 1
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00:01:19,816 --> 00:01:20,944
what happens is the following:
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00:01:20,944 --> 00:01:23,172
Well, you get a negative sign and you
39
00:01:23,172 --> 00:01:25,060
get the sum. And now what you've done is
40
00:01:25,060 --> 00:01:27,092
pulled down a factor of 'i'. So, now all
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00:01:27,092 --> 00:01:36,630
of a sudden, you have the same infinite
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00:01:36,630 --> 00:01:38,055
sum you have in the other equation
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00:01:38,055 --> 00:01:39,919
appearing. This sum here you can't do
44
00:01:39,919 --> 00:01:42,284
immediately, by the geometric series trick
45
00:01:42,284 --> 00:01:44,383
because instead of it being 1 plus 'p'
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00:01:44,383 --> 00:01:45,540
plus 'p' squared plus 'p' to the
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00:01:45,540 --> 00:01:47,681
fourth...plus 'p' cubed and so forth,
48
00:01:47,681 --> 00:01:49,388
it now has these funny terms in the front
49
00:01:49,388 --> 00:01:50,620
here hanging off. But notice that if you
50
00:01:50,620 --> 00:01:52,974
the derivative of the partition function,
51
00:01:52,974 --> 00:01:54,663
you pull down that factor of 'i'.
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00:01:54,663 --> 00:02:00,053
So, in fact, we can now rewrite this
53
00:02:00,053 --> 00:02:01,827
second constrained equation here as
54
00:02:01,827 --> 00:02:08,308
1 over 'z', minus, 'd' 'Z' 'd' lambda 1
55
00:02:08,308 --> 00:02:11,041
equals 4.
56
00:02:11,041 --> 00:02:12,422
So, first thing you have to do is take
57
00:02:12,422 --> 00:02:13,241
a derivative of 'z' with respect to
58
00:02:13,241 --> 00:02:14,654
lambda 1 and then we have to divide
59
00:02:14,654 --> 00:02:16,538
by 'z'. And so we can do that quite
60
00:02:16,538 --> 00:02:18,438
easily. We have a factor 'e' to the
61
00:02:18,438 --> 00:02:20,606
negative lambda 1 on the top, 1 minus 'e'
62
00:02:20,606 --> 00:02:23,189
to the negative lambda 1 squared - that's
63
00:02:23,189 --> 00:02:25,242
the derivative of 'z' with respect
64
00:02:25,242 --> 00:02:28,408
to lambda 1...minus sign. And then we have
65
00:02:28,408 --> 00:02:31,358
a factor of 1 over 'z', and all that does
66
00:02:31,358 --> 00:02:33,708
is pull one of these out. So 'z' is 1 over
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00:02:33,708 --> 00:02:36,274
1 minus 'e' to the negative lambda 1. Ok?
68
00:02:36,274 --> 00:02:38,627
So if we divide by that...we just lose
69
00:02:38,627 --> 00:02:44,849
one of these. And so now, we've turned
70
00:02:44,849 --> 00:02:45,918
this equation here, which we would have
71
00:02:45,918 --> 00:02:47,440
trouble solving, because it's an infinite
72
00:02:47,440 --> 00:02:49,396
sum, and you know, putting an infinite sum
73
00:02:49,396 --> 00:02:50,872
into MatLab tends to take a long time to
74
00:02:50,872 --> 00:02:54,846
solve. But instead, we have a analytic
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00:02:54,846 --> 00:02:57,538
form, a functional form that we can put in
76
00:02:57,538 --> 00:03:01,208
right away. Ok? So now, all we have to do
77
00:03:01,208 --> 00:03:03,318
is solve this equation here. Find the
78
00:03:03,318 --> 00:03:06,483
value of lambda 1 that sets this term here
79
00:03:06,483 --> 00:03:09,036
equal to 4. Once we find that value of
80
00:03:09,036 --> 00:03:11,903
lambda 1, we can then plug into 'z'
81
00:03:11,903 --> 00:03:14,322
here, ok? And once we know both 'z' and
82
00:03:14,322 --> 00:03:18,173
lambda 1, we can recover not just the
83
00:03:18,173 --> 00:03:21,424
functional form of the probability of
84
00:03:21,424 --> 00:03:28,254
waiting for 'x', but actually we'll be
85
00:03:28,254 --> 00:03:31,634
able to compute the waiting time as a
86
00:03:31,634 --> 00:03:33,654
function of 'x', ok? Er, the probability
87
00:03:33,654 --> 00:03:36,949
of a waiting time for a particular value,
88
00:03:36,949 --> 00:03:39,000
ok? So, we're left with this equation here
89
00:03:39,000 --> 00:03:41,319
and instead of solving that exactly,
90
00:03:41,319 --> 00:03:43,304
because we have some logarithms, I can
91
00:03:43,304 --> 00:03:46,372
tell you the answer that lambda 1 is equal
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00:03:46,372 --> 00:03:54,001
to roughly 0.22. And from there, we can
93
00:03:54,001 --> 00:04:00,224
also compute the value of 'z', and we can
94
00:04:00,224 --> 00:04:08,578
now write out the proportionality,
95
00:04:08,578 --> 00:04:12,727
roughly speaking, ok? So this distribution
96
00:04:12,727 --> 00:04:19,535
here, with appropriate normalization,
97
00:04:19,535 --> 00:04:23,144
will be the exponential distribution, it's
98
00:04:23,144 --> 00:04:24,818
an exponential distribution, linear in
99
00:04:24,818 --> 00:04:26,544
the waiting time 'x'. And what I've shown
100
00:04:26,544 --> 00:04:28,972
to you, is this distribution has the
101
00:04:28,972 --> 00:04:31,444
following properties, ok? It satisfies
102
00:04:31,444 --> 00:04:35,017
this linear constraint. It's normalized.
103
00:04:35,017 --> 00:04:38,628
Ok? And it has the maximum entropy of all
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00:04:38,628 --> 00:04:40,448
the distributions with those two previous
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properties.