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All right. Big day today. We're going to talk about – we're going to do our final

application of convolution. I suppose I shouldn't say "final application of convolution”

because it is the kind of operation that comes up repeatedly throughout the course. But

sort of as the – as a last treatment of the kind of areas we've been talking about, I wanna

talk about application convolution to the central limit theorem.

And this is one of my favorite topics because it just is so – it's such an important result,

and it's such, in some sense, a surprising application of convolution. And the result itself

is just so – I don't know. It has this air of mystery about it that it's a real – I think it's a

real treat to see how it's – see how they – see how these ideas play out.

So I wanna talk about convolution and the central limit theorem. I will describe this, but

it actually takes a little while to set up. We have to develop the appropriate language to

get a precise statement of what the result is, and then understand how we're going to

approach it. But this – the central limit theorem is, I say, one of the cornerstones of

probability. And it's not only very important from a theoretical point of view from the

development probability, but it's extremely important from the practical point of view. It

has to do with the ubiquity of the bell-shaped curve, or why is it that so many things are

distributed according to a Gaussian.

The central limit theorem, which, like all good things, has a three-letter acronym that

goes along with it – the CLT somehow explains the universal – explains the bell-shaped

curve – the universal appearance of the bell-shaped curve, that is to say the Gaussian in

probability.

And there is a quote, actually, on this. There's a quote that's often repeated and

connection with the central limit theorem. I put it in the notes, and I just wanted to read it

to you. It's by G. Litmann, who's a French physicist.

He says something like, "Everyone believes in the normal approximation."

The normal approximation is another word for the Gaussian approximation for the bellshaped

curve.

Says, "Everyone believes in the normal approximation, the mathematicians because they

think that with a," – excuse me – "the experimenters because they think it is a

mathematical theorem, the mathematicians because they think it is an experimental fact.

It's got something for everyone."

Everybody buys it. Everybody believes it. And today, you will know why it's true.

Well, what does it say? Well, again, it's gonna take us a little while – a couple of

iterations – before we get to a precise statement. What it says is, it says that most

probabilities – I mean, this is sort of an intuitive or informal way of putting it. Most

probabilities – some kind of average, really, is the way of thinking about it – in some

kind of average sense are calculated or approximated, at least, as if they were distributed

– as if they were determined by a bell-shaped curve – by a Gaussian.

The key word here, as it turns out in the precise understanding of this, is averaging. In an

average sense – or averaging many outcomes or many outcomes contributing in an

average sense to the final outcome are distributed according to a Gaussian.

Now, the picture of this, so – before I make that precise, the picture is something like

this. If you have a Gaussian, we're going to be working with a number of standard

Gaussians.

application of convolution. I suppose I shouldn't say "final application of convolution”

because it is the kind of operation that comes up repeatedly throughout the course. But

sort of as the – as a last treatment of the kind of areas we've been talking about, I wanna

talk about application convolution to the central limit theorem.

And this is one of my favorite topics because it just is so – it's such an important result,

and it's such, in some sense, a surprising application of convolution. And the result itself

is just so – I don't know. It has this air of mystery about it that it's a real – I think it's a

real treat to see how it's – see how they – see how these ideas play out.

So I wanna talk about convolution and the central limit theorem. I will describe this, but

it actually takes a little while to set up. We have to develop the appropriate language to

get a precise statement of what the result is, and then understand how we're going to

approach it. But this – the central limit theorem is, I say, one of the cornerstones of

probability. And it's not only very important from a theoretical point of view from the

development probability, but it's extremely important from the practical point of view. It

has to do with the ubiquity of the bell-shaped curve, or why is it that so many things are

distributed according to a Gaussian.

The central limit theorem, which, like all good things, has a three-letter acronym that

goes along with it – the CLT somehow explains the universal – explains the bell-shaped

curve – the universal appearance of the bell-shaped curve, that is to say the Gaussian in

probability.

And there is a quote, actually, on this. There's a quote that's often repeated and

connection with the central limit theorem. I put it in the notes, and I just wanted to read it

to you. It's by G. Litmann, who's a French physicist.

He says something like, "Everyone believes in the normal approximation."

The normal approximation is another word for the Gaussian approximation for the bellshaped

curve.

Says, "Everyone believes in the normal approximation, the mathematicians because they

think that with a," – excuse me – "the experimenters because they think it is a

mathematical theorem, the mathematicians because they think it is an experimental fact.

It's got something for everyone."

Everybody buys it. Everybody believes it. And today, you will know why it's true.

Well, what does it say? Well, again, it's gonna take us a little while – a couple of

iterations – before we get to a precise statement. What it says is, it says that most

probabilities – I mean, this is sort of an intuitive or informal way of putting it. Most

probabilities – some kind of average, really, is the way of thinking about it – in some

kind of average sense are calculated or approximated, at least, as if they were distributed

– as if they were determined by a bell-shaped curve – by a Gaussian.

The key word here, as it turns out in the precise understanding of this, is averaging. In an

average sense – or averaging many outcomes or many outcomes contributing in an

average sense to the final outcome are distributed according to a Gaussian.

Now, the picture of this, so – before I make that precise, the picture is something like

this. If you have a Gaussian, we're going to be working with a number of standard

Gaussians.

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