1
00:00:01,860 --> 00:00:03,680
In this video, we're
going to establish

2
00:00:03,680 --> 00:00:06,400
a nice property of
the Poisson process.

3
00:00:06,400 --> 00:00:07,860
Here is our setting.

4
00:00:07,860 --> 00:00:12,920
We have a Poisson process
with arrival rate, lambda,

5
00:00:12,920 --> 00:00:15,360
and so arrivals keep coming.

6
00:00:15,360 --> 00:00:21,530
And we watch this process
until a certain random time, T.

7
00:00:21,530 --> 00:00:24,770
So T is this time here.

8
00:00:24,770 --> 00:00:29,190
Now, T is an exponential random
variable with some parameter,

9
00:00:29,190 --> 00:00:32,800
and T is independent from
the Poisson arrival process.

10
00:00:32,800 --> 00:00:35,350
What we're interested
in is the number

11
00:00:35,350 --> 00:00:39,240
of arrivals that happened
during this time.

12
00:00:39,240 --> 00:00:41,270
How can we answer this question?

13
00:00:41,270 --> 00:00:42,250
There are two ways.

14
00:00:42,250 --> 00:00:44,810
One is using mathematical
manipulations.

15
00:00:44,810 --> 00:00:47,190
The other is using intuition.

16
00:00:47,190 --> 00:00:49,750
Let's see what it
would take if we wanted

17
00:00:49,750 --> 00:00:52,760
to solve the problem
using formulas.

18
00:00:52,760 --> 00:00:59,770
So let's call N-T, with capital
T, the number of arrivals

19
00:00:59,770 --> 00:01:02,800
until time T in our
Poisson process.

20
00:01:02,800 --> 00:01:06,490
And we wish to find the
distribution of N-T. So we

21
00:01:06,490 --> 00:01:08,690
want to calculate, for
example, the probability

22
00:01:08,690 --> 00:01:11,455
that N-T is equal to
a specific number, k.

23
00:01:14,110 --> 00:01:18,330
Now, we do not know very much
about this random variable,

24
00:01:18,330 --> 00:01:24,170
but we do know the probability
of the random variable, N-T,

25
00:01:24,170 --> 00:01:27,130
If we have a deterministic time.

26
00:01:27,130 --> 00:01:30,490
So perhaps we can
condition by fixing

27
00:01:30,490 --> 00:01:34,450
the value of the random
variable, capital T-- that

28
00:01:34,450 --> 00:01:37,350
is, to condition on
that random variable,

29
00:01:37,350 --> 00:01:41,530
taking on a specific value.

30
00:01:41,530 --> 00:01:43,220
What happens in this case?

31
00:01:43,220 --> 00:01:46,520
Well, if I tell you that
capital T is equal to little t,

32
00:01:46,520 --> 00:01:50,729
this probability is going to
be the same as the probability

33
00:01:50,729 --> 00:01:57,570
that N with little t is equal
to k, where N with little t

34
00:01:57,570 --> 00:02:01,040
is the number of arrivals
until time, little t.

35
00:02:01,040 --> 00:02:03,770
But the number of arrivals,
until a certain time,

36
00:02:03,770 --> 00:02:05,730
is a Poisson random variable.

37
00:02:05,730 --> 00:02:09,949
So we do know what
this probability is.

38
00:02:09,949 --> 00:02:13,470
It is lambda to the k, e
to the [minus] lambda t,

39
00:02:13,470 --> 00:02:17,180
divided by k factorial.

40
00:02:17,180 --> 00:02:19,670
Now, if we have this
conditional probability, how can

41
00:02:19,670 --> 00:02:23,150
we get the unconditional
probability?

42
00:02:23,150 --> 00:02:26,540
This is done by using the
total probability theorem.

43
00:02:26,540 --> 00:02:30,260
We consider all possible
values of little t, which

44
00:02:30,260 --> 00:02:33,440
are all the numbers
from 0 to infinity.

45
00:02:33,440 --> 00:02:38,070
We weigh each possible
value of little t

46
00:02:38,070 --> 00:02:43,130
according to the corresponding
PDF of the random variable, T.

47
00:02:43,130 --> 00:02:46,900
And we have this equality.

48
00:02:46,900 --> 00:02:48,640
We know what this term is.

49
00:02:48,640 --> 00:02:49,565
It is this expression.

50
00:02:53,130 --> 00:02:56,320
And the density of
capital T, since it

51
00:02:56,320 --> 00:03:01,260
is an exponential variable,
the density takes this form.

52
00:03:01,260 --> 00:03:03,550
And so, in order to
find the distribution

53
00:03:03,550 --> 00:03:05,640
of the random
variable, N capital T,

54
00:03:05,640 --> 00:03:09,230
all that we need to do is
to calculate this integral.

55
00:03:09,230 --> 00:03:12,090
But this is a rather
messy integral.

56
00:03:12,090 --> 00:03:14,410
So let us try to
see if we can find

57
00:03:14,410 --> 00:03:16,250
a shortcut to this problem.

58
00:03:19,590 --> 00:03:22,570
So here is what we have.

59
00:03:22,570 --> 00:03:26,960
We have a Poisson process.

60
00:03:26,960 --> 00:03:29,770
Let's call it the
first Poisson process,

61
00:03:29,770 --> 00:03:32,470
that has a certain rate, lambda.

62
00:03:32,470 --> 00:03:36,620
And this Poisson process has
arrivals at various times.

63
00:03:40,310 --> 00:03:45,340
And then we have a random
variable, capital T,

64
00:03:45,340 --> 00:03:47,860
which is exponential.

65
00:03:47,860 --> 00:03:52,460
How should we think about an
exponential random variable?

66
00:03:52,460 --> 00:03:54,970
We can think of an
exponential random variable

67
00:03:54,970 --> 00:03:59,560
as being the first arrival
in some Poisson process.

68
00:03:59,560 --> 00:04:04,120
So let us put down a second
Poisson process with rate mu.

69
00:04:04,120 --> 00:04:06,480
Since we have assumed
that capital T is

70
00:04:06,480 --> 00:04:09,530
independent from the
red Poisson process,

71
00:04:09,530 --> 00:04:11,920
we can just assume
that this blue Poisson

72
00:04:11,920 --> 00:04:13,925
process is independent
from the first.

73
00:04:17,649 --> 00:04:23,020
Now, let us merge
the two processes.

74
00:04:23,020 --> 00:04:27,070
And we're going to form
a merged process that

75
00:04:27,070 --> 00:04:31,660
records an arrival at
those times at which either

76
00:04:31,660 --> 00:04:33,650
of the two processes
have an arrival.

77
00:04:37,500 --> 00:04:39,040
This is the time of interest.

78
00:04:41,950 --> 00:04:44,320
And the random variable
that we're interested in

79
00:04:44,320 --> 00:04:47,430
is the number of red
arrivals until that time.

80
00:04:47,430 --> 00:04:53,010
That random variable we call
N capital T. The discussion

81
00:04:53,010 --> 00:04:56,800
will be a little simpler if we
define another random variable,

82
00:04:56,800 --> 00:05:02,500
K, which is N capital T plus 1.

83
00:05:02,500 --> 00:05:07,360
So K is the number of
arrivals in the merged process

84
00:05:07,360 --> 00:05:08,760
until this time.

85
00:05:08,760 --> 00:05:11,320
That is, we take those
arrivals of the red process,

86
00:05:11,320 --> 00:05:14,660
and we also include
that arrival here.

87
00:05:14,660 --> 00:05:18,240
So if the number of
arrivals that we got here

88
00:05:18,240 --> 00:05:26,290
was N capital T, here we have
arrivals 1, 2, and so on.

89
00:05:26,290 --> 00:05:30,930
And this is arrival number
K. So K is a random variable,

90
00:05:30,930 --> 00:05:33,260
and we want to find what it is.

91
00:05:33,260 --> 00:05:36,230
So what are we asking?

92
00:05:36,230 --> 00:05:39,000
What is K?

93
00:05:39,000 --> 00:05:53,580
K is the number of arrivals
in the merged process

94
00:05:53,580 --> 00:05:58,320
until you get an arrival
in the merged process which

95
00:05:58,320 --> 00:06:00,195
is coming from the blue process.

96
00:06:13,870 --> 00:06:18,460
And now, here is how we can
think of this situation.

97
00:06:18,460 --> 00:06:29,335
Think of each arrival in the
merged process as a trial.

98
00:06:35,860 --> 00:06:38,130
Each one of these
arrivals is coming

99
00:06:38,130 --> 00:06:41,950
either from the red process
or from the blue process.

100
00:06:41,950 --> 00:06:49,710
Let us call it a success if it
comes from the blue process.

101
00:06:58,190 --> 00:07:08,000
So in that case, K is
the number of trials

102
00:07:08,000 --> 00:07:09,935
until we have a success.

103
00:07:14,220 --> 00:07:17,930
K is the number of arrivals
in the merged process

104
00:07:17,930 --> 00:07:19,910
until we have a
successful arrival,

105
00:07:19,910 --> 00:07:24,290
meaning one that came
out of the blue process.

106
00:07:24,290 --> 00:07:27,070
So what do we know
about those trials?

107
00:07:27,070 --> 00:07:30,270
What are their
statistical properties?

108
00:07:30,270 --> 00:07:32,830
First, what is the
probability of success?

109
00:07:37,580 --> 00:07:40,050
The probability of
success is the probability

110
00:07:40,050 --> 00:07:43,060
that when you get an arrival
in the merged process,

111
00:07:43,060 --> 00:07:49,580
it is coming from
the blue process.

112
00:07:49,580 --> 00:07:54,460
And as we have seen, given an
arrival in the merged process,

113
00:07:54,460 --> 00:07:56,450
there is probability
that it's coming

114
00:07:56,450 --> 00:07:58,150
from this particular
process that's

115
00:07:58,150 --> 00:08:02,050
proportional to the arrival
rate of that particular process.

116
00:08:02,050 --> 00:08:05,420
And it is equal, as
we have discussed

117
00:08:05,420 --> 00:08:09,560
before, it is equal to this.

118
00:08:09,560 --> 00:08:11,890
In addition, we have
discussed that when

119
00:08:11,890 --> 00:08:13,720
you look at the merged
process, and you

120
00:08:13,720 --> 00:08:16,390
ask, what was the
origin of that arrival,

121
00:08:16,390 --> 00:08:19,540
and you ask, what was the
origin of that arrival,

122
00:08:19,540 --> 00:08:21,750
the answers to
these two questions

123
00:08:21,750 --> 00:08:24,360
are independent of each other.

124
00:08:24,360 --> 00:08:26,470
In other words, the
origin of this arrival

125
00:08:26,470 --> 00:08:29,060
is statistically
independent from the origin

126
00:08:29,060 --> 00:08:30,710
of that arrival.

127
00:08:30,710 --> 00:08:35,924
So this means that these
trials are independent.

128
00:08:39,760 --> 00:08:44,560
So what we're looking
at is a random variable,

129
00:08:44,560 --> 00:08:48,330
which is the number of trials
until the first success,

130
00:08:48,330 --> 00:08:51,070
in a sequence of
independent trials,

131
00:08:51,070 --> 00:08:54,090
where each trial has
a success probability

132
00:08:54,090 --> 00:08:55,800
equal to that value.

133
00:08:55,800 --> 00:08:59,065
And we know what
that distribution is.

134
00:08:59,065 --> 00:09:04,000
It is a geometric with
this particular parameter.

135
00:09:04,000 --> 00:09:05,900
This gives us the
probability distribution

136
00:09:05,900 --> 00:09:09,970
of the random variable, K, and
once you have the PMF of K,

137
00:09:09,970 --> 00:09:13,890
you just shift it by
1 to the left in order

138
00:09:13,890 --> 00:09:16,380
to get the probability
distribution, the PMF,

139
00:09:16,380 --> 00:09:19,960
of the random
variable, N capital T.

140
00:09:19,960 --> 00:09:22,770
And so this is the
answer to this problem.

141
00:09:22,770 --> 00:09:28,520
The number of arrivals during an
exponentially distributed time

142
00:09:28,520 --> 00:09:32,030
interval [has] a
geometric distribution

143
00:09:32,030 --> 00:09:35,640
that's shifted by 1 to the left.