WEBVTT
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we're told the family are bell shaped curve or described
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by dysfunction. And this is called a normal density
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function and theirs to constantly have them. You,
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which is described demean. And we have, Ah
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, another continent called Sigma. And that tells us
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two standard deviations. And we're going to be looking
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at the case where knew is you could zero giving
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us this more simple function and we're being asked a
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couple of things are being told to find the Athen
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toe maxim by inflection point. So the story starts
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. So to find the ascent of this function we're
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looking for when x goes tio positive or negative infinity
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. And so if you look at this function is
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e to the minus X wherever treason, we're square
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. So this could be re written as why you
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call one over, um e to the X squared
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over two to sigma squared over to think much word
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. So if you look at this as X goes
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to really, really big numbers say e to the
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one hundred over two Sigma squared sigma still positive.
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So it's going to eat in a one hundred or
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eat it, you get one over a really big
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numbers to get one over infinity. Well, that's
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just really, like gets closer to zero. So
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and since exponential functions since it's being squared, it
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doesn't matter if he's going to positive or a negative
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infinity. Ill struggle approaches zero. So it will
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have a horizontal ascent or that Y equals zero.
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And then we're told, then we're being asked to
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find the maximum value. So this means by taking
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the first derivative. So if we take the first
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driven and we're gonna have a problem, Max,
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and come down to B minus e ah, negative
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X squared over two Sigma squared, sometimes to act
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. So there's a time to act over two Sigma
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squared and said This equals zero and you can see
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that this from every calls, you know. So
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if you plug in X equals your you get zero
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so you get X equals zero and you plug in
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fo zero. You get one. So we have
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maximum local Mac one. And in this case it
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is the absolute max, too. And then we're
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told, find the inflection point. It involves taking
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a second to evidence and this is a bit more
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work. So this is going to come out to
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be minus a to the minus. X squared over
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two Sigma squared. That's a signal Times one over
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Sigma squared Truth X over Sigma squared. Um,
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times e to the minus X squared that's squared over
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two. Take my squared Our time's too x time
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to act over two sigma squared And then you can
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pull out a common factor of the, um minus
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X squared. So you e to the minus x
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squared over two sigma squared And you can also pull
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out a factor of I think, Mr Ford.
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And then you'LL get X square minus sigma squared because
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zero and you can't This cannot go zero. So
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we set experiments Tecnicos Sigma squared equals zero. So
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this gives us X square equal Sigma squared Take the
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square root you get X eagles off plus or minus
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sigma So that's the inflection point in collection point.
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And then we're being asked to evaluate what role the
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signal play in the shape of our curve. So
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he looked at this function again, so we had
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remember the function with y equals one over Run E
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to the X squared over two Sigma squared and is
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being asked to look at how signal Please rule.
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So as signal home and the value of sigma gets
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closer to zero. But this number gets really literally
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closest zero you get, you get X squared Reminder
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. I'm just looking writing up the exponents of X
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squared over to signal square. So as the denominator
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gets really, really small, you get a really
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, really big numbers. Sigma approaches, you know
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, you get a big number. So you get
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a really, really, really, really, really
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big number. And the fact that as he goes
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really close to infinity so you get one over e
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Teo the infinity again. They got one of the
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gates infinity and remember, one over a really big
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number approaches zero. So sigma is close to zero
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. Um, what's happening, toe? Why is
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that? Why is approaching zero? So signal my
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girls to deal. Why approaches why is also getting
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closer and closer? It is, you know.
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And so what happens is is this makes this are
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the function itself gets a lot closer to the Y
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axis, so it becomes more and more narrow and
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more more closer and compacted together. And the improvised
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versa. So a Sigma Gesche. So stigma approaches
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. Um, another signal gets bigger, so it's
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just a signal. Goes to infinity was just a
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significant infinity. Um, X squared over two things
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squared. Well, what happens to this? So
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Sigma Squared goes to really, really big number.
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So you get get X squared over Sigma squared and
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this approaches. It's approaches zero and then you get
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and then zero. And so what's happening? Teo
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it It's the opposite. It's going to happen.
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So a signal goes closer and closer. Why Asset
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Purchase one And this will become a bit more clear
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when I showed the graph, so actually have taken
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the liberty of drawing from the function before. So
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, as you can see, the green curve right
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here is AA minus X square over four. So
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if I increase it, that's their increasing to sixteen
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. This purple graph shows that is much more widespread
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and much more further out, which is what we
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predicted. And as they get closer and closer to
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a smaller number of the denominator, you get a
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more more sophisticated, more compacted graph, and that's
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exactly what we predicted.