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PROFESSOR: So as an
aside, for those of you

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who have an understanding
of probability

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theory and stochastic
processes, let

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me just explain why it is valid
to bound the main transmission

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rate in defining the indoor
reproductive number when

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we might be more
concerned about bounding

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the probability
of a transmission.

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So let's let a capital T
be a random variable which

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describes the random
number of transmissions

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that occur in the room.

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And this is specifically for
the case for one infector

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or infected person and
n minus 1 susceptible.

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So again, the situation of
the reproductive number where

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for every person that
comes in we want to know

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is there going to
be transmission,

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and typically, only one
infected person would be seen.

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So this is a random variable,
and let's let ft of n

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be the probability
density function that

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gives the probability of
little n transmissions.

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And then we can define the risk
of a transmission, the risk

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of at least one transmission,
as the probability

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that this random variable
takes on a value, which

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is greater than or equal to 1.

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OK.

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Well, in terms of the
probability density function

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then, that would be a sum from
1 to infinity of the ft event,

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basically.

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So we're just adding
the probabilities

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that we're not seeing a
transmission to those--

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or possible transition to those
different numbers of people.

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Now I'd like to do
a little calculation

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to get an upper bound
on this quantity.

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So we can say that
this is less than

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or equal to the
sum from n equals 1

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to infinity of n ft of n.

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Now, this is just a
mathematical trick here.

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So n refers to the natural
numbers 1, 2, 3, 4, et cetera.

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Those are all positive
numbers and they're

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all greater than or equal to 1.

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So if I take 1 in this
expression over here

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and replace it
with little n, I'm

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only increasing the value
of that sum, because also,

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this ft is a probability
density that has to be positive.

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And then now I can also
say that this is actually

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equal to throwing in n equals
0, because that is a term that

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is actually identically 0.

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So I can change the summation.

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And then by definition
here, this thing

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is the expected value of
the number of transmissions,

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because I'm summing the
number of transmissions

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little n times the probability
of that event occurring.

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So that is the definition
of the average.

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So what we're seeing here is
that the risk of a transmission

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is rigorously bounded
above by the expected

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number of transmissions.

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And so therefore, if we require
that this is less than epsilon,

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our risk tolerance that
we've just introduced,

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this is a conservative bound
on the true risk, which let's

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say here is defined by rt.

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So if your goal is to control
the probability of having

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at least one
transmission, so basically

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to ensure that no
transmissions occur,

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then you would do well to
bound the expected number

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of transmissions, because
that's an upper bound.

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It can also be shown
that as epsilon

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goes to 0, so we're talking
about very low probabilities

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of transmission, and
oftentimes that is the case,

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then rt is asymptotically
the same as the expected

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number of transmissions.

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So this overall risk of a
transmission and the expected

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number are the same.

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And in fact, that's
one way to understand

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sort of even the
definition of a probability

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in terms of an expected
number of events.

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And we are typically
thinking of cases where

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epsilon is much less than 1.

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So for those of you that have
some background in probability,

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you may recognize that
what I've just done here

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is an example of a much
more general result, which

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is called Markov's Inequality.

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So now we can safely
proceed by continuing

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to work with average values of
all the quantities of interest.