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Let me introduce you to
normal distribution,

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which is one of the most commonly used
distributions in statistical analysis,

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and even in everyday conversations.

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A large body of academic,
scholarly and professional work rests

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on the assumption that the underlying
data follows a normal distribution.

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The defining characteristics of normal
distribution is this bell shaped curve

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which you're familiar
with from your textbooks.

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Mathematically, normal distribution
is presented by this equation.

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We can say that the normal distribution
relies on three inputs and

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f stands for functions, so
function of x, mu and sigma.

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X is your data, x is a random variable and
it can attain any reasonable value.

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Mu stands for the mean.

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And sigma is standard deviation.

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And the mathematical
formulation is right here,

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which is 1 divided by sigma times and

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then square root of 2 times pi,
pi is 3.142 or 22/7.

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And then you have the exponential here.

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Do not forget this minus sign so
exponential of this entity

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which is minus and
in the numerator x- mean or

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x- mu d hold squared divided
by 2 times sigma squared.

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So let me explain.

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1 divided by the standard deviation and

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then square root of 2 times pi,
2 is known and pi's known.

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So is the value for exponential and

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what is not known is the sigma
which you'd obtained from the data,

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that is standard deviation and the mean,
which is also coming from data.

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So you have the mean and
the standard deviation, and

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x is the random variable who's mean and
standard deviation you're using.

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You put this all together and
then you get the normal distribution,

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the bell shaped curve
that you saw earlier.

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I also introduce you to
the standard normal.

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And standard normal is when we say that
x is a variable that has a mean 0 and

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standard deviation of 1.

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So what's mean 0 and
standard deviation 1 look like?

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If you replace mu with 0 and
standard deviation or sigma with 1,

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the equation reduce to this
entity which is 1 / 2 times pi.

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Notice the sigma here which have great
out a bit so that it doesn't interfere.

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Sigma is 1 so one times anything would
be the same, so I've removed this.

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And then e to the power -(x- mu),
but because mu is 0 so

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x- 0 is x, so
((x)2/2) times sigma squared, sigma,

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remember is 1,
the square of 1 is 1, so 2 times 1.

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So removing sigma because it's 1 and

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anything multiplied with
1 is the same entity.

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So how do I generate this
normal density or a bell curve?

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Let's assume that the underlying
variable has a mean 0 and

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the stand deviation of 1 and
x varies between -4 and 4.

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So the mean is 0 and the minimum
value is -4, the maximum value is 4.

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And I would substitute these -4
to 4 values in this equation.

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This is the only thing that's changing,
the x here is the only entity that is

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changing, and let's see if this could
generate the standard normal curve.

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>> Let us do this in Python,
we'll use the matplotlib function which

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you are already familiar with for
the graphics, NumPy library,

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as well as the norm.pdf function
in the SciPy stats library.

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In this example,
I have used increments of 0.1.

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This will generate the standard normal
curve that you hear about in statistics.

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