FRM: Normal probability distribution
Summary
TLDRDavid Harper from the Bionic Turtle offers an insightful tutorial on the normal distribution, a fundamental concept in probability. He explains the distribution's characteristics, including its bell curve shape, symmetry, and the two parameters defining it: mean and standard deviation. Harper also discusses the misuse of the normal distribution in financial asset returns due to their fat-tailed nature and positive skew, contrasting it with the normal distribution's thin tails. He concludes by highlighting the distribution's prevalence due to its simplicity and the central limit theorem.
Takeaways
- 📚 The normal distribution, also known as the bell curve or Gaussian distribution, is the most common and familiar probability distribution.
- 📈 It's often used to describe asset returns, despite being a misuse due to its symmetrical and clean nature compared to the actual fat-tailed distributions of financial returns.
- 📊 The normal distribution is characterized by its symmetry and the presence of two parameters: mean and standard deviation, which fully describe the distribution.
- 📉 The script illustrates the mismatch between the normal distribution and actual stock returns, such as Google's, which are positively skewed and have fatter tails.
- 🔢 The normal distribution's density function is implemented with the formula involving π and e, and parameters for mean and standard deviation.
- 📝 The script demonstrates how to plot the normal distribution using both an Excel function and a manual formula, showing their equivalence.
- 🔧 By adjusting the mean and standard deviation, the script shows how the normal distribution curve shifts without changing its shape, highlighting the distribution's behavior.
- 🌐 The normal distribution is fully described by only two parameters, making it convenient for various applications.
- 📋 The script explains the '68-95-99.7' rule, which describes the proportion of the area under the normal distribution curve within one, two, and three standard deviations from the mean.
- 📘 The normal distribution is continuous, symmetrical, and has a kurtosis of three, indicating no excess kurtosis or fat tails.
- 🌟 The prevalence of the normal distribution is due to its simplicity and the central limit theorem, which states that the sampling mean of samples tends to be normally distributed, emphasizing its importance in statistics rather than in risk management.
Q & A
What is the normal distribution also known as?
-The normal distribution is also known as the normal bell curve and occasionally referred to as a Gaussian distribution.
Why is the normal distribution sometimes misused to describe asset returns?
-The normal distribution is sometimes misused to describe asset returns due to its convenience and elegant properties, despite being inappropriate for financial returns because it does not accurately represent their characteristics.
What does the histogram of Google's stock price daily returns show about the fit of the normal distribution?
-The histogram of Google's stock price daily returns shows that the normal distribution does not fit well, as there is a positive skew and the presence of outlier returns, indicating fat-tailed distributions.
What is meant by the term 'leptokurtosis' in the context of asset returns?
-Leptokurtosis refers to the phenomenon where asset returns have fatter tails than the normal distribution, meaning there is a higher likelihood of extreme values occurring.
What are the two parameters needed for the normal distribution function?
-The two parameters needed for the normal distribution function are the mean and the standard deviation, which determine the peak and the dispersion of the curve, respectively.
What is the standard normal distribution and what are its parameters?
-The standard normal distribution is a specific case of the normal distribution with a mean of zero and a standard deviation of one.
How does changing the mean affect the normal distribution curve?
-Changing the mean shifts the peak of the normal distribution curve along the x-axis without altering the shape of the curve.
How does changing the standard deviation affect the normal distribution curve?
-Changing the standard deviation affects the width of the normal distribution curve, with a larger standard deviation resulting in a wider, more dispersed curve.
What are the 'rules of thumb' regarding the area under the normal distribution curve?
-The rules of thumb state that approximately 68% of the area under the curve is within one standard deviation of the mean, 95.5% within two standard deviations, and 99.7% within three standard deviations.
What are the characteristics of the normal distribution?
-The characteristics of the normal distribution include being continuous, fully described by two parameters (mean and standard deviation), symmetrical with a skew of zero, and having no excess kurtosis (kurtosis of three).
Why is the normal distribution so common in statistics?
-The normal distribution is common due to its simplicity with only two parameters and the central limit theorem, which states that the sampling mean of samples tends to become normally distributed.
Outlines
📊 Introduction to the Normal Distribution
David Harper introduces the normal distribution, commonly known as the bell curve or Gaussian distribution, emphasizing its prevalence in probability theory. He explains the misuse of the normal distribution in financial asset returns, using Google's stock price as an example to illustrate the positive skew and fat-tailed nature of actual returns. Harper demonstrates the normal distribution's symmetrical and clean curve, contrasting it with the lepto kurtosis of financial returns, and introduces the normal distribution function and its parameters: mean and standard deviation.
📚 Characteristics and Properties of the Normal Distribution
This paragraph delves into the characteristics of the normal distribution, highlighting its full description by only two parameters: the mean and standard deviation. Harper discusses the implications of changing these parameters on the distribution's curve, emphasizing the distribution's behavior and the concept of lepto kurtosis. He also outlines the empirical rules of thumb related to the area under the normal curve, such as the percentages of area covered by one, two, and three standard deviations. Harper concludes by explaining why the normal distribution is so common, attributing it to its simplicity and the central limit theorem, which states that the sampling mean of samples tends to become normally distributed.
Mindmap
Keywords
💡Normal Distribution
💡Bell Curve
💡Gaussian Distribution
💡Asset Returns
💡Histogram
💡Skewness
💡Leptokurtosis
💡Volatility
💡Standard Deviation
💡Cumulative Distribution Function (CDF)
💡Central Limit Theorem
💡Kurtosis
Highlights
The normal distribution, also known as the bell curve or Gaussian distribution, is the most common and familiar probability distribution.
Despite its misuse, the normal distribution is often used to describe asset returns due to its convenient properties.
Financial returns, such as Google's stock, typically do not match the normal distribution due to positive skew and fat tails.
The normal distribution function is represented by a formula involving pi and the natural number e.
The normal distribution is characterized by two parameters: the mean and the standard deviation.
Excel's built-in function 'NORM.DIST' can be used to calculate the normal distribution.
Changing the mean and standard deviation in the normal distribution alters the curve's position and dispersion but not its shape.
The normal distribution is fully described by only two parameters, making it convenient for use.
Rules of thumb for the normal distribution include percentages of area under the curve within one, two, and three standard deviations.
The normal distribution is continuous, symmetrical, and has a kurtosis of three, indicating no excess kurtosis or fat tails.
The prevalence of the normal distribution is due to its simplicity and the central limit theorem, which states that the sampling mean of samples tends to be normally distributed.
The central limit theorem does not necessarily imply special significance in risk management, as it pertains to central tendency rather than tail risks.
The normal distribution's properties make it a popular choice despite its limitations in accurately representing financial asset returns.
David Harper of the Bionic Turtle provides a tutorial on the normal distribution, emphasizing its characteristics and common applications.
The tutorial includes a practical demonstration using Google's stock returns to illustrate the mismatch with the normal distribution.
The video explains the mathematical formula for the normal distribution and its implementation in Excel.
The tutorial concludes by discussing the limitations of the normal distribution in risk metrics and the importance of understanding its properties.
Transcripts
hello this is David Harper of the Bionic
turtle with a brief tutorial on the
normal distribution easily the most
common popular and familiar probability
distribution of all probability
distributions the normal is also called
the normal bell curve and occasionally
we hear it referred to as a Gaussian
distribution that's Gaussian with a G a
Gaussian distribution is the same thing
as a normal it's just a little more
technical term okay to show you the
normal curve let me show you in practice
oftentimes we'll use it to describe
asset returns even as we use it we
typically know that it's a misuse that
it's sort of inappropriate to financial
returns but it's so convenient that
sometimes we'll use it as sheer
convenience it has elegant and
convenient properties but just to show
you why it's a misuse here I pulled
periodic returns for Google's stock
price this ending mid December of 2007
so these are the daily periodic returns
that I've then plotted onto a histogram
or frequency chart and so you might say
it first glance well that looks like the
normal bell curve and if I collected
more data I would definitely get a
smoother distribution but if I
superimpose the normal density function
so here's that bell curve plotted in
blue if it's for most financial asset
returns I'm not going to get a good
match so it Google's pretty typical here
in the sense that there's not a match
you can see the normal curve is
symmetrical and clean whereas my Google
returns well first of all they are
positively skewed see how there's a
positive skew over to the right leaving
the space here secondly what's harder to
see is Google has something because I
have a fat thick a thick blue line here
Google has some real outlier returns
here on the positive side the actual
asset returns are typically fat-tailed
they have fatter tails technically
that's called lepto kurtosis so now let
me take a closer look at the normal
distribution function
density function so here it is plotted
the plot is an implementation here of
this
formula which has pi in it as well as
the natural e that has a value about 2.7
1 in addition to that really there are
embedded the two parameters that we need
for the normal distribution here's
volatility that's that measure of
dispersion or that second moment measure
we need that
and here's Z a standardized variable
that is a function of the mean and the
standard deviation so embedded in the Z
we've got the mean and the volatility
those are the two parameters we need for
the standard deviation I've got a mean
here input and I've got a standard
deviation here of one so I have any
numbers in here I'm going to go ahead
and put in standard values the standard
normal distribution by definition when I
say standard it means there's a mean of
zero and a standard deviation of one
such that my X values here equal my
standard variables and then I can plot
this normal distribution this is a
density function also called probability
mass function here over to the right so
as you can see I've done that two ways
first here I use the built in excel
function equals norm dist takes four
parameters first here I've got b7 first
it takes the x value so that's the value
of x on the x-axis it takes the mean
which I've got a zero
it takes the standard deviation I've got
a 1 and then it takes a true if we want
a cumulative distribution or a false if
we want the density function which we do
want so that's my function that's my
straight use of the excel function and
then just to prove to you that the
formula is correct here instead of a
function I've got a formula and that
formula right here matches this so it
gives me Y as a function of X the
standard variable here the in this case
for this first point it's a negative 4
so that negative 4 is my Z right
here having that and then also accessing
my also using as an input my standard
deviation I solve for y which is then
going to be that probability here so
that's the Y access value here a Y value
here here's here's the formula which
implements this which is equivalent to
the built-in function and that gives me
none
I can now I've got a mean of 0 here but
if you watch this chart I'm going to
just change the mean to 10 for example
and the curve doesn't change at all the
curves very well behaved that way but
the mean changes to 10 and now I'm going
to change the standard deviation to 5
for example so watch the chart well ok
it's hard to see the x axis changes but
this the x axis change so the dispersion
became greater and i'll change the
standard deviation back to 1 but I can
change the mean of the standard
deviation the point of this that I
wanted to show you is that I only change
those two values one of the
characteristics of the normal
distribution is that it's fully
described by only two parameters the
mean that sets this that sets the peak
of the curve here and the what I've been
calling interchangeably the volatility
of the standard deviation that's the
measure of dispersion or scatter that
tells me how wide this is that's it
aside from that this tendency of the
tails to be skinny relative to real
returns that's just a weakness or flaw
of the normal so another characteristic
is we have these rules of thumb which
you may have seen before that refer to
area under the curve so if I just look
back at this this is the density
function the cumulative function is
implied by the area under the curve so
if we talk about how much area under the
curve goes plus or minus one standard
deviation to the left and right well
then we have these rules of thumb about
68% of the area under the curve is
covered by plus or minus one standard
deviation about 95 and a half percent is
covered by two standard deviations about
99
seven is covered by three standard
deviations so those are rules of thumb
that tell us how much of the area under
the curve is covered by the distance
plus or minus one two and three standard
deviations respectively so finally
before I finish let me just explain the
properties of the normal distribution
first it's continuous it's not discrete
notice this line is solid not dotted
technically speaking if I want to check
for the probability of value I have to
check an interval because there are an
infinite number of points on this line
that would be as opposed to a discrete
variable or discrete distribution such
as the binomial or purse on as I
mentioned second characteristic there's
only two parameters fully described by
mean and volatility or standard
deviation this makes it very convenient
and handy to use many distributions are
more complex require more parameters and
therefore subject to well just more
model risk or input variation third we
say the normal is symmetrical it has a
or technically it has a skew of zero
fourth we say there is no excess
kurtosis the there are no fat tails the
kurtosis therefore on the normal is
going to be three so we say the kurtosis
is three or the excess kurtosis is zero
same thing why is it so common well
there are really two reasons and I would
argue there are only two reasons there
is nothing inherently special about the
normal distribution it is not
necessarily appropriate for risk metrics
and nothing elevates at any special
importance in the study of risk simply
because risk is concerned with the tail
the strength of the normal is about the
central tendency the two reasons is
common are one it's easy due to these
parameters and two this is very profound
the central limit theorem the central
limit theorem says the sampling mean of
the samples tends toward become normal
normally distributed well that is a
statement about the central tendency of
the normal it's not really a statement
about the tails and the proteins of the
tails therefore it's not necessarily
doesn't necessarily have any special
significance in the context of risk but
the central limit theorem does explain
why it's so prevalent as a distribution
so this is David Harper of the monic
turtle thanks for your time
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