WEBVTT
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Language: en
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Now I move into the fourth example, consider
a repairman who replaces a light bulb the
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instant it burns out. Suppose, the first light
bulb is put in at time zero and let X suffix
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i either be the lifetime of i-th light bulb
okay.
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You define the random variable Tn is a sum
of n Xi’s where Xi’s are iid random variables.
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Xi with a lifetime of the ith light bulb and
when Xi are iid random variable you are defining
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Tn is a X1 plus X2 plus Xn and so.
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So the Tn be the time the nth light bulb burns
out because the Tn is a X1 plus X2 and so
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on till Xn therefore Tn be the time the nth
light bulb burns out. Assume that Xi’s is
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exponential distribution with a parameter
lambda. We know that already Xi’s are iid
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random variable, now I am making the further
assumption, Xi’s follows exponential distribution
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with a parameter lambda. That means, you know
what is the mean of this random variable.
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Since it is exponential distribution with
the parameter lambda, this becomes 1 divided
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by lambda. Also one can use the result, Tn
by n that is nothing but 1 divided by n summation
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of Xi where i is running from 1 to n as n
tends to infinity one can prove Tn by n tends
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to 1 divided by lambda that is a mean of the
random variable Xi almost surely.
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I am not proving here the way you do the sequence
of random variable converges to another random
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variable converges takes place in probability
or in distribution or in rth mean or almost
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surely one can prove this the Tn by n converges
to 1 by lambda almost surely. That means we
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can conclude the random variable X1, X2 and
so on obeys strong law of large numbers.
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Because the Tn by n that is nothing but 1
by n of summation of Xi that converges to
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the value 1 by lambda almost surely we can
conclude the sequence of random variable Xi
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obeys a strong law of large number. Even though
in this problem, I made the assumption Xi’s
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follows the exponential distribution with
the parameter lambda, in general the lifetime
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can be any distribution. So this problem will
be discussed in detail in renewal processes.
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So as such here we are making the assumption
of distribution of Xi is exponential distribution
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therefore I made it converges takes place
almost surely to the value 1 by lambda, this
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can be generalised. There are many more problems
of the similar kind but we are discussing
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only few problems, therefore we can use the
similar logic of finding the moment generating
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function, then concluding the distribution
and finding the limiting distribution.
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Or you verify whether the sequence of random
variables converges takes place in mean, converges
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takes place in probability or converges takes
place in distribution, or converges in the
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rth mean or converges almost surely. This
can be used in any problem of at the same
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way what I have done it here. And I have not
discussed any problem in the central limit
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theorem but that will be used many times therefore
I have not given any problems for the central
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limit theorem.
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Let X1, X2, so on be a sequence of random
variables, each having student t distribution
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with n degrees of freedom. Our interest is
to find out the limiting distribution of a
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student t distribution. We know that the probability
density function of f of x for the random
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variable Xn is given by gamma of a n plus
1 by 2 divided by square root of n times pi
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multiplied by gamma of n by 2 multiplied by
1 plus x square by n power minus n plus 1
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by 2.
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So this is the probability density function
of a random variable Xn.
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Our interest is to find out the limiting distribution
of the random variable Xn. For larger, for
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large n, we have the results limit as n tends
to infinity of gamma of n plus 1 by 2 divided
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by square root of n pi of gamma of n by 2
is 1 divided by square root of 2 pi using
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Stirling’s approximation. And also, limit
n tends to infinity of 1 plus x square by
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2 the whole power minus n plus 1 by 2 that
we know, that is e power minus x square by
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2.
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Hence, the limit n tends to infinity of the
probability density function of the random
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variable Xn becomes 1 divided by square root
of 2 pi e power minus x square by 2. Since,
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right hand side is the probability density
function of a standard normal distribution,
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we conclude for a larger n the sequence of
random variables X1, X2, Xn and so on that
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tends to the random variable Z and this converges
takes place in distribution where Z is standard
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normal distribution.
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So this is a simple example of the sequence
of random variables each having a student
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t distribution, the limiting distribution
converges to standard normal and that converges
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in distribution.