We now move to the world of Continuous Random Variables,
and we will talk about the Continuous Uniform Distribution,
the Normal Distribution, applications of the Normal
Distribution, and finally the Lognormal Distribution.
Let us consider the Continuous Uniform Distribution,
First the Continuous Uniform Distribution is defined
over a range that spans between some lower limit
"a" and some upper limit "b". Let us say that we have a random variable
which is uniformly distributed between 5 and 10.
The Continuous Uniform Distribution will simply
look like a straight line. It has uniform that is
why we have a straight line, what that means is that
the random variable will be between 5 and 6 so this,
the probability is denoted by this area over here.
That probability will be the same as the probability
of the random variable falling between 6 and 7, and
so on. The overall probability that the random variable
falls between 5 and 10 is equal to one, that means that
the total area of this rectangle is one. So the probability
that the random variable falls between 5 and 6 is
equal to this rectangle, the smaller rectangle
over here, essentially, the bar. That probability is
simply 1/5. The probability of the random variable
between 6 and 7 is 1/5 and so on, if you add all these
five numbers so 5 to 6, 6 to 7, 7 to 8, 8 to 9, and 9 to 10. You
should get the total probability of one. The next
question is what is the height of this rectangle?
We have already established that if the rectangle
base is going from 5 to 10 so the width is 5. We've established
that the area is 1 so remember from your basic mathematics
that area is base into height. If the area is 1 and
the base is 5 then the height must be 1/5. That 5 is essentially
b - a which in our example is 10 - 5. So this is very
straightforward. If I asked you what is the probability
that the random variable is less than 7 then essentially
I'm asking what is the probability of this rectangle
because anything less than 5 has a probability of
0. The probability that the random variable is less
than 7 is simply 2/5 so that is also the area of this
rectangle. So this is fairly straightforward, once
you understand what is going on. Let us look at an example. Random
variable X follows a continuous uniform distribution
between 10 and 20. What is the probability of an outcome
between 15 and 25? So all you do is recognize the fact
that a random variable is between 10 and 20 and
we want to know what is the probability of being between
15 and 25. If we are to draw this out we need to recognize that
the base is equal to 10 so 20 - 10 is 10. We know that
the area, total area, is going to be 1. So what is the
height? The height is simply 1/10 because 1/10 * 10
is going to be 1. The question now is what is the probability
that the random variable is greater than 15? So that
essentially is this shaded rectangle. There is
nothing over here because this probability is zero.
So the probability of being between 15 and 25 is the
same as a probability of being between 5 and 20. Now
you can simply see that this is half of the total area.
So the probability is 5/10, which is 0.5. Again, I'll
just emphasize that this is fairly straightforward,
and one other point that you should recognize is
that when we have a continuous uniform distribution,
we always speak in terms of a range. So the probability
of an outcome between 15 and 20 or between 15 and 15.5
so there is always a range. The reason is that the probability of a specific number so
probability of 15 is going to be essentially 0, and given
that there are infinite number of possibilities
between 10 and 20, the probability of one specific
number is effectively zero. So that's why there
is always a range that we talk about in the context
of continuous distributions or continuous random
variables. We are still on the continuous uniform
distribution. But now we will look at a Cumulative
Distribution. If you recall from our earlier discussion,
when we say cumulative, we are essentially saying
that the probability of a random variable being
less than or equal to a certain number. If we continue
with the example we just talked about, we are going
from 10 to 20. So the cumulative distribution would
essentially be P(X ≤ a specific
value of the random variable). If I say P( X ≤
15) that is the cumulative probability up
to 15 so that means all the probability that we
accumulate up to this point in time. So that cumulative
probability is equal to 1/2 and we just saw that. The
way you can say this is, the cummulative probability up to this
point in time is zero. So if you bring this down over
here at the bottom, we have the cumulative probability
so up to point 10. So random variable is equal to 5, the
cumulative probability is 0. Even at 10, the cumulative
probability is zero, but as we start increasing
beyond that then the cumulative probability starts
rising because we are accumulating all this probability.
When we are at 15, which is the midpoint then the cumulative
probability will be 0.5 as you get to 20 the cumulative
probability becomes 1. After that as you keep increasing,
the cumulative probability stays at 1 because here
the probability is zero so you are not accumulating
anymore probability. Cumulative probability is
also expressed using the letter F. So if F(15), in
our example, is the probability that the random variable
is less than or equal to 15. So that is 1/2 F(20)
is equal to 1, F(100) so anything more than 20,
would also be equal to 1. Let's look at this practice
question now. The execution time for a single order
is uniformly distributed between 10 and 18 minutes.
If an order is placed at 2:30 p.m. What is the probability
that the order will execute after 2:45? So here is
how you can do this. Let us say that we are currently
at time zero. So the order is placed at 2:30. Now, the
order will take place between 10 and 18 minutes and
we have a uniform distribution. So that means that
the order will execute somewhere between 2:40 and
2:48. The question is what is the probability that
the order will execute after 2:45. So if you simply
look at this. We are asking for this probability, that is
simply equal to 3. This is 3 in the numerator divided
by the total distance, which is 8, so the probability
is 3 / 8.