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Now, when working with probability
distributions, we have the probability

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function, we have the probability curve.
There's also something known as the

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cumulative distribution function. The
cumulative distribution function or the

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CDF of a random variable is denoted by
capital F. This just represents the

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probability that the random value is less
than or equal to a particular value. So,

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f(x), capital F(x) is equal to the
probability that big X is less than or

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equal to little x. So, it's the area under
the probability curve, if you have a

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continuous random variable to the left of
x, okay? If you have a discreet

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distribution, then you just count up the
probability values for all x less than or

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equal to a particular value and you
usually get a stair step kind of function.

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The cumulative distribution function
satisfies the following properties. So, if

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x2 is bigger than x1, then the area to the
left of x2 is bigger than the area to the

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left of x1. The CDF at minus infinity is
zero. The area to the left of minus

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infinity is zero. And the CDF at infinity
is equal to one, so the area to the left

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of plus infinity has to be equal to one.
This is just says the total area in the

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probability curve is equal to one. Because
the total area of the probability curve is

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equal to one, the probability that x is
greater than, the random variable is

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greater than or equal to x is one minus
the probability that it's less than or

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equal to x. If we want to know the area,
or if we want to know the probability that

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x is between, that the random variable is
between x1 and x2, it's the CDF evaluated

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x2 - the CDF evaluated at x1. So, this
result here is one of the main reasons why

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we look at the cumulative distribution
function. If we want to know the

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probability over an interval, we can just
calculate the difference between the CDF

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values. And so, if we have some function
that can compute the CDF, then we can

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compute the probabilities. And if we have
a continuous random variable, it turns out

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that the probability curve, the density is
equal to t he derivative of the CDF. So,

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there's a result from Calculus. And this
is just an example of Fubini's theorem.

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So, let's look at an example. Suppose we
have a uniform distribution over zero

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actually, before we go to the uniform
distribution, let's just look at so, this

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discrete distribution for Microsoft stock
that we had before. So, if we look at this

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distribution here. So, this is the
probability mass function. And if we

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wanted to know, what is the cumulative
distribution? What is the probability that

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x is less than or equal to zero? Well,
it's the sum of the probabilities that x

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is less than zero. So, we calculate the
cumulative distribution function, we see

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the probability that x is less than -four
is zero. The probability that the return

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is less than -three is, is just the
probability that it's, it's equal to -30%.

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So, this is the probability that you're in
a depression. And then the probability

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that the return is less than say -ten%.
Well again, that's the sum of these two

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events. And then, we just add up the
probabilities and we get this stair step.

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So, that's a discreet distribution. We
have a continuous distribution, say the,

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say the uniform distribution on 0,1. So,
in that case, we have a very simple

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distribution. So, we have a distribution
that is essentially the square, that's,

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that's supposed to be one. And so, we wan
to know what is the probability that this

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random variable x is less than or equal to
little x? Well, that's the area under the

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curve to the left of x. So we, think of
some value here, x. Then this area on to

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the probability curve to the left of x is
just the shaded area here, which I'll

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shade in. And so, this represents the
probability that x is less than or equal

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to x. It's just the area under the
probability curve to the left of X. And we

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can do this integration very easily cuz
the probability curve is just equal to

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one. The, this, this probability is just
the interval of the z from zero to x, so

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that's equal to X. So, for uniform
distribution on the unit square, if we

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want to calcul ate the CDF is just a
straight line that looks like this. So,

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this is F(x) of x is equal to x is the
CDF. So, this is the case where we can

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analytically derive what this CDF is by
actually evaluating the interval. And if

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we want to find the probability, say, that
x is between zero and a half, then what is

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this? This is just the area under the
probability curve between zero and a half.

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We can evaluate the CDF at a half minus
the CDF at zero. The CDF at a half, using

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this result, is a half. The CDF at zero is
zero, so the area between zero and a half

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is equal to a half, okay? And then
similarly, the result from Calculus,

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because the CDF is equal to x, if we take
the derivative of x, we get one, and

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that's exactly the probability curve. So,
the continuous random variable, we

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integrate, we take the derivative of the
CDF, we get the, the probability curve.

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All right, so, I want to make a remark
here. So in this example, let's say, when

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we had this the probability curve on the
unit square, right, we can think of the

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probability, say, that this is the
probability, say, between 0.5 and 0.6 is

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the area under the curve, so this is equal
to the area under the curve, okay? So,

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what happens if we consider the
probability as you know, this, this

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interval gets smaller and smaller and
smaller. So, the idea is we want to think

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of, what's the probability at a single
point? So, if so, if we, we squish this

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rectangle such that we just get a single
point, what's the area associated at a

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single point? It's zero. So, with a
continuous random variable, the

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probability that a continuous random
variable is exactly equal to a particular

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point is, is zero. Now, that's a little
counter intuitive, right? Because I, cuz

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when you see the probability of, of x
being equal to a particular point is zero,

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it's like saying that it can never happen,
right? That's not true. And so, it' s sort

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of like with a continuous random variable,
again, there's an infinite number of

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values between zero and one, right? And
think about the idea of guessing, you kno

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w, exactly what x is going to happen,
right? You can have as many decimal points

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after this, you know, the accuracy that
you want. So, in some respects you know,

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probability of a single point for
continuous random variables just doesn't

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make any sense. And so, for continuous
random variables, we define probabilities

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over intervals and not at distinct points.
And with the continuous, and because of

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this result, we have a continuous random
variable. The probability that x is less

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or equal to x is the same as the
probability that x is strictly less than

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x, because the probability at a single
point is equal to zero.