In this lecture we're going to talk
about continuous variables, so we've
talked about discrete variables up until
now. Discrete variables can only take on
distinct values, but continuous variables
can take on any value. So with continuous
variables we don't have a notion of
probabilities for exact values because X
can take on an infinite number of values,
so the probability of equaling any
specific exact value is zero. We can have
probabilities for ranges though. So, for
example, we can say the probability of X
being between 3.13 and 3.15 is greater
than zero. We have a useful function
called the cumulative distribution
function, or the CDF, that helps us
measure such probabilities. We usually
label this function as big F of X, and so
the definition of F(X) is it's the
probability that the random variable big
X is greater than negative infinity, but
less than little x. Note that the
probability of big X being between
negative infinity and positive infinity
is 1 since X has to take on a value,
therefore the value of big F of positive
infinity is equal to 1. Now how about
going back to our original problem if we
want to calculate the probability that X
is between 3.13 and 3.15. That would just
be big F of 3.15 minus big F of 3.13. So
now let's talk about the other useful
function when we're talking about
continuous variables. This one's called
the probability
density function, or the PDF. We
usually denote it by little f of X, and
it is defined as the derivative of big F
of X with respect to X, so it's like the
slope of big F of X. Note that this
function can be greater than one since
it's not a probability, it is a
probability density f of X, little f of X
does have to be greater than or equal to
0 though. So here's one example where
little f of X can be bigger than one. So,
let's say little f of X is uniform
between zero and 0.1, so that means if
you try to sample from this random
variable X you'll always get a value
between zero and 0.1, and the probability
of any particular value is equal to all
the others. Now I'm going to claim that
little f of x has to equal 10 if X is
between 0 and 0.1, and 0
otherwise. Now why is this, because big F
of X. Since little f of X is the
derivative of big F of X, big F of X is
the integral of little f of X. In the
integral we can take the constant out
and then calculate the integral from 0
to X. Now we know that from above big F
of infinity has to equal 1, so the integral
from minus infinity to infinity equals
to 1, but since little f of x is 0 after
0.1. we can just take the integral from
0 to 0.1. That gives us
0.1c, and if we solve for c,
c equals 10. Therefore, we've seen a
scenario where little f of X can have a
value greater than one because it's a
probability density, and not a probability
value. Later on in this course we'll
look at more complex continuous
distributions.