Now something else that we care about.
In, in a distribution.
Is known as a quantile.
And quantiles are related to the.
Cumulative distribution function.
So, let's consider a random variable.
That's a continuous random variable.
That has a, a continuous cumulative
distribution function.
So we have.
Capital F of X is the probability that,
that big X is less or equal to little X.
Alright?
So the definition of the.
Quantile of a distribution so we take some
probability level aplha and so the alpha,
an alpha sum number between zero and one
so the alpha times a 100 percent quantile
distribution is the value Q of alpha such
that the CDF evaluated the quantile is
equal to alpha.
Now the best way of thinking about this is
to draw a picture.
So the idea here, if this is the
probability curve.
Say F of X and let's specify some
probability here, alpha.
Say alpha is equal five%.
Then, the five percent quantile.
Is the value of x such that the
probability to the left of the quantile is
equal to this, probability value, okay.
So quantiles are the value in the sample
space such that the probability to the
left of quantile is equal to your
specified alpha.
Okay, so, so, that's again, the definition
of a quanta, we think we look at through
here.
Now with the continuous random variable.
So go back and Let's think of, X(f) of X,
for a continuous random variable.
The, cube of distribution function.
Looks like this.
And so X is over here, so here's minus
infinity.
Here's infinity over here.
And if we have some value X, then the CDF,
here, and if we is this height, so if this
is we put a ax scale on this, this is
zero, this is one, then This is FX of
little x.
That's the probability that big X is less
or equal to little x.
So when we have a continuous distribution
the area under the probability curve we
can write the cumulative distribution
function.
It looks like this s-shape function.
And so, for any value of X we can figure
out by going to the distribution function
what, what this probability is.
Then we think of quantiles.
We can think of some probability value,
like.05, and then we can say, given the
distribution function, given my
probability value, what value of X gives
me this probability?
So that, again, that defines what the
quantile is.
So, when we are working with a CDF we
take'X' and then we get a probability
value, and when we work with quantiles we
go in the other direction.
We take a probability value and then we
figure out what the quantile is.
So in this way we got f of x.
And then this way so we end up we do whats
called the inverse function.
So we start from here and go down then
were inverting this CDF function given the
probability to get a quantile.
So, the picture that I just described.
So, if this, if we have a continuous,
cumulative distribution function, then the
inverse of the distribution function
exists.
And then we can solve for the quantile by
applying the inverse function to, the
random variable.
Now, the inverse function satisfies FX
inverse of FX of Q.
Is equal to Q alpha.
This is equal to Q alpha.
So we apply the inverse function to the,
the CDF at, at a point and that negates
it.
So think of like, the exponential
logarithmic function.
The exponential is the inverse of the
logarithmic function and vice versa.
So either logarithm of x is equal to x.
So the same thing is going on here, we
think of this is as the CDF and the
inverse CDF is the function such that it
satisfies this property.
So, if we can mathematically compute the
inverse function, then we can get the
quantile.
So, that's why this inverse CDF is often
referred to as the quantile function for,
a distribution.
Now, particular quantiles that we're
interested in.
So the one percent quantile is QO1, the
five percent quantile and the 50 percent
quantile is known as the median.
So again, if we think of the, of a
distribution.
We plot it's probability curve.
And if this area's one%.
This is the one percent quantile.
The area's five%.
This is the five percent quantile.
And then the median is the 50 percent
quantile, that's such that the area is to
the left of that.
Now we're gonna find in finance that
quantiles are particularly interesting
things because they, they tell us about.
The probability of loss.
So if we think of.
The random variable.
Associated with this probability curve as
a rate of return.
So then, and values over here would be
negative rates of return.
And values over here would be positive
rates of return.
So this low, this lower quantile
represents a small negative number that
happens with a, you know, a particular
probability.
So when we talk about, you know, with one
percent probability, how much could we
lose?
Well, it's equal to the one percent
quantile of the return distribution.
And, we can say with five percent
probability, how much could we lose.
Then, we're concerned with the five
percent quantile of the distribution.
So, these quantiles, particularly when we
have distributions that represent rates of
return, tell us about, the probabilities
associated with losing money.
And, so those are, that's a very
important, very interesting way of
thinking about risk.
Now, sometimes we can figure out the
quantile function, for a particular random
variable.
So for example, if we have the uniform
distribution, we know that the, CDF.
Looks like this.
So.
The inverse distribution function.
Is exactly the same.
So for a, uniform random variable if I
tell you the probability I get X.
Or if I take, take X I give you.
Get the same probability value.
So the inverse function.
Quantile function.
Satisfies this property.