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Now something else that we care about.
In, in a distribution.

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Is known as a quantile.
And quantiles are related to the.

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Cumulative distribution function.
So, let's consider a random variable.

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That's a continuous random variable.
That has a, a continuous cumulative

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distribution function.
So we have.

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Capital F of X is the probability that,
that big X is less or equal to little X.

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Alright?
So the definition of the.

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Quantile of a distribution so we take some
probability level aplha and so the alpha,

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an alpha sum number between zero and one
so the alpha times a 100 percent quantile

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distribution is the value Q of alpha such
that the CDF evaluated the quantile is

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equal to alpha.
Now the best way of thinking about this is

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to draw a picture.
So the idea here, if this is the

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probability curve.
Say F of X and let's specify some

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probability here, alpha.
Say alpha is equal five%.

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Then, the five percent quantile.
Is the value of x such that the

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probability to the left of the quantile is
equal to this, probability value, okay.

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So quantiles are the value in the sample
space such that the probability to the

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left of quantile is equal to your
specified alpha.

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Okay, so, so, that's again, the definition
of a quanta, we think we look at through

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here.
Now with the continuous random variable.

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So go back and Let's think of, X(f) of X,
for a continuous random variable.

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The, cube of distribution function.
Looks like this.

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And so X is over here, so here's minus
infinity.

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Here's infinity over here.
And if we have some value X, then the CDF,

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here, and if we is this height, so if this
is we put a ax scale on this, this is

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zero, this is one, then This is FX of
little x.

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That's the probability that big X is less
or equal to little x.

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So when we have a continuous distribution
the area under the probability curve we

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can write the cumulative distribution
function.

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It looks like this s-shape function.
And so, for any value of X we can figure

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out by going to the distribution function
what, what this probability is.

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Then we think of quantiles.
We can think of some probability value,

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like.05, and then we can say, given the
distribution function, given my

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probability value, what value of X gives
me this probability?

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So that, again, that defines what the
quantile is.

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So, when we are working with a CDF we
take'X' and then we get a probability

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value, and when we work with quantiles we
go in the other direction.

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We take a probability value and then we
figure out what the quantile is.

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So in this way we got f of x.
And then this way so we end up we do whats

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called the inverse function.
So we start from here and go down then

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were inverting this CDF function given the
probability to get a quantile.

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So, the picture that I just described.
So, if this, if we have a continuous,

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cumulative distribution function, then the
inverse of the distribution function

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exists.
And then we can solve for the quantile by

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applying the inverse function to, the
random variable.

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Now, the inverse function satisfies FX
inverse of FX of Q.

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Is equal to Q alpha.
This is equal to Q alpha.

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So we apply the inverse function to the,
the CDF at, at a point and that negates

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it.
So think of like, the exponential

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logarithmic function.
The exponential is the inverse of the

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logarithmic function and vice versa.
So either logarithm of x is equal to x.

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So the same thing is going on here, we
think of this is as the CDF and the

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inverse CDF is the function such that it
satisfies this property.

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So, if we can mathematically compute the
inverse function, then we can get the

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quantile.
So, that's why this inverse CDF is often

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referred to as the quantile function for,
a distribution.

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Now, particular quantiles that we're
interested in.

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So the one percent quantile is QO1, the
five percent quantile and the 50 percent

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quantile is known as the median.
So again, if we think of the, of a

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distribution.
We plot it's probability curve.

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And if this area's one%.
This is the one percent quantile.

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The area's five%.
This is the five percent quantile.

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And then the median is the 50 percent
quantile, that's such that the area is to

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the left of that.
Now we're gonna find in finance that

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quantiles are particularly interesting
things because they, they tell us about.

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The probability of loss.
So if we think of.

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The random variable.
Associated with this probability curve as

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a rate of return.
So then, and values over here would be

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negative rates of return.
And values over here would be positive

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rates of return.
So this low, this lower quantile

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represents a small negative number that
happens with a, you know, a particular

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probability.
So when we talk about, you know, with one

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percent probability, how much could we
lose?

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Well, it's equal to the one percent
quantile of the return distribution.

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00:06:46,630 --> 00:06:50,394
And, we can say with five percent
probability, how much could we lose.

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Then, we're concerned with the five
percent quantile of the distribution.

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So, these quantiles, particularly when we
have distributions that represent rates of

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return, tell us about, the probabilities
associated with losing money.

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00:07:04,259 --> 00:07:08,525
And, so those are, that's a very
important, very interesting way of

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thinking about risk.
Now, sometimes we can figure out the

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00:07:16,324 --> 00:07:21,371
quantile function, for a particular random
variable.

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So for example, if we have the uniform
distribution, we know that the, CDF.

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00:07:28,920 --> 00:07:30,302
Looks like this.
So.

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The inverse distribution function.
Is exactly the same.

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00:07:34,448 --> 00:07:39,612
So for a, uniform random variable if I
tell you the probability I get X.

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00:07:39,612 --> 00:07:44,122
Or if I take, take X I give you.
Get the same probability value.

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So the inverse function.
Quantile function.

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Satisfies this property.