Let us now give an example of a continuous random variable--
the uniform random variable.
It is patterned after the discrete random variable.
Similar to the discrete case, we will have a range of
possible values.
In the discrete case, these values would be the integers
between a and b.
In the continuous case, any real number between a and b
will be possible.
In the discrete case, these values were equally likely.
In the continuous case, at all points, we have the same
height for the probability density function.
And as a consequence, if we take two intervals that have
the same length, then these two intervals will be assigned
the same probability.
Intuitively, uniform random variables model
the following situation.
We know that the numerical value of the random variable
will be between a and b.
But we know nothing more.
We have no reason to believe that certain locations are
more likely than others.
And in this sense, the uniform random variable models a
situation of complete ignorance.
By the way, since probabilities must add to 1,
the area of this rectangle must be equal to 1.
And therefore, the height of this rectangle has to be 1
over b minus a, so that we have a height of 1
over b minus a.
We have a length of b minus a.
So the product of the two, which is the
area, is equal to 1.
Finally, here's a more general PDF, which
is piecewise constant.
One thing to notice is that this, in particular, tells us
that PDFs do not have to be continuous functions.
They can have discontinuities.
Of course, for this to be a legitimate PDF, the total area
under the curve, which is the sum of the areas of the
rectangles that we have here, must be equal to 1.
With a piecewise constant PDF, we can calculate probabilities
of events fairly easy.
For example, if you wish to find the probability of this
particular interval, which is going to be the area under the
curve, that area really consists of two pieces.
We find the areas of these two rectangles, add them up, and
this gives us the total probability of
this particular interval.
So at this point, our agenda, moving
forward, will be twofold.
First, we will introduce some interesting
continuous random variables.
We just started with the presentation of the uniform
random variable.
And then, we will also go over all of the concepts and
results that we have developed for discrete random variables
and develop them again for their continuous counterparts.