2.3 Continuous Probability Distributions
========================================

:bolditalic:`Key Terms`

:bolditalic:`Probability density function（機率密度函數）`
   A function whose area over an interval gives probability.

:bolditalic:`PDF（機率密度函數）`
   Short form of probability density function.

:bolditalic:`Density（密度）`
   Probability concentration represented by height; probability itself is represented by area.

:bolditalic:`Cumulative distribution function（累積分配函數）`
   A function that gives the probability that a random variable is less than or equal to a given value.

:bolditalic:`CDF（累積分配函數）`
   Short form of cumulative distribution function.

:bolditalic:`Interval probability（區間機率）`
   Probability that a continuous random variable falls within an interval.


2.3.2 Learning Outcomes
-----------------------

After completing this section, students should be able to:

- explain why continuous probability is described by a PDF rather than a PMF;
- explain why :math:`P(X=x)=0` for a continuous random variable;
- interpret probability as area under a density curve;
- identify the conditions of a valid PDF;
- calculate simple interval probabilities from a PDF;
- construct a CDF from a simple continuous PDF;
- use :math:`F(x)` to calculate interval probabilities;
- explain the relationship between a PDF and a CDF;
- explain why including or excluding interval endpoints does not change a continuous probability.


2.3.3 From PMF to PDF
---------------------

In the previous section, a discrete random variable was described by a
**probability mass function**:

.. math::

   p(x)=P(X=x).

A discrete random variable can place positive probability at individual
values.

A continuous random variable is different. Its possible values fill an
interval, so probability is not assigned to individual points. Instead, a
continuous distribution is described by a **probability density function**,
or PDF.

.. image:: ../_static/figures/random_variables/figure_2_3_1_pmf_vs_pdf.png
   :width: 78%
   :align: center
   :alt: Comparison of probability mass at discrete points with probability density spread continuously across an interval.

For a continuous random variable,

.. math::

   P(X=x)=0

for every individual value :math:`x`.

This does not mean that :math:`x` is impossible. A single point has zero
width, so it contributes zero area under the density curve.

.. note::

   **PMF versus PDF**

   For a discrete random variable,

   .. math::

      p(x)=P(X=x).

   For a continuous random variable,

   .. math::

      f(x)\neq P(X=x).

   Probability is obtained from **area**, not from the density height at one
   point.


2.3.4 A Single Example: Bus Waiting Time
----------------------------------------

Let :math:`X` be the waiting time, in minutes, for the next bus.

Suppose the waiting time can be anywhere from 0 to 10 minutes and is modeled
with the constant density

.. math::

   f(x)=
   \begin{cases}
   \frac{1}{10}, & 0\leq x\leq10,\\
   0, & \text{otherwise}.
   \end{cases}

This one example will be used to introduce both the PDF and the CDF.


2.3.5 Probability Density Function
----------------------------------

A valid PDF must satisfy two conditions:

.. math::

   f(x)\geq0

for all :math:`x`, and

.. math::

   \int_{-\infty}^{\infty}f(x)\,dx=1.

For the waiting-time example, the total area is

.. math::

   10\left(\frac{1}{10}\right)=1,

so the density is valid.

Probability over an interval is the area under the PDF:

.. math::

   P(a<X<b)
   =
   \int_a^b f(x)\,dx.

For example,

.. math::

   P(2<X<6)
   =
   \int_2^6 \frac{1}{10}\,dx
   =
   \frac{4}{10}
   =
   0.40.

The corresponding area is shown below.

.. image:: ../_static/figures/random_variables/figure_2_3_2_pdf_area_probability.png
   :width: 62%
   :align: center
   :alt: Constant probability density from zero to ten with the area between two and six shaded.

The height :math:`1/10` is a **density**. The probability :math:`0.40` comes
from height multiplied by interval width.


2.3.6 Height Is Not Probability
-------------------------------

For a continuous distribution,

.. math::

   f(x)\neq P(X=x).

In the waiting-time example,

.. math::

   f(4)=\frac{1}{10},

but

.. math::

   P(X=4)=0.

The density height describes how probability is concentrated near a value.
Probability is obtained only after an interval has been specified.

A density value can even be greater than 1. This is allowed as long as the
total area under the PDF is exactly 1.


2.3.7 Endpoints Do Not Matter
-----------------------------

For a continuous random variable,

.. math::

   P(X=a)=P(X=b)=0.

Therefore,

.. math::

   P(a<X<b)
   =
   P(a\leq X<b)
   =
   P(a<X\leq b)
   =
   P(a\leq X\leq b).

For example,

.. math::

   P(2<X<6)
   =
   P(2\leq X\leq6)
   =
   0.40.

The endpoints do not change the area and therefore do not change the
probability.


2.3.8 Cumulative Distribution Function
--------------------------------------

The CDF is defined for both discrete and continuous random variables:

.. math::

   F(x)=P(X\leq x).

For a continuous random variable, the CDF is accumulated area under the PDF:

.. math::

   F(x)
   =
   \int_{-\infty}^{x}f(t)\,dt.

For the waiting-time example,

.. math::

   F(x)=
   \begin{cases}
   0, & x<0,\\
   \frac{x}{10}, & 0\leq x\leq10,\\
   1, & x>10.
   \end{cases}

Thus,

.. math::

   F(2)=0.20,
   \qquad
   F(6)=0.60.

The CDF is shown below.

.. image:: ../_static/figures/random_variables/figure_2_3_3_continuous_cdf.png
   :width: 62%
   :align: center
   :alt: Continuous cumulative distribution function for a waiting time uniformly distributed between zero and ten minutes.

Unlike the discrete CDF, this CDF increases continuously rather than by
jumps.


2.3.9 Finding Probabilities from the CDF
-----------------------------------------

The CDF provides accumulated probability from the left.

For example,

.. math::

   P(X\leq6)=F(6)=0.60.

For an interval,

.. math::

   P(a<X<b)=F(b)-F(a).

Therefore,

.. math::

   P(2<X<6)
   =
   F(6)-F(2)
   =
   0.60-0.20
   =
   0.40.

This is the same probability obtained earlier from the area under the PDF.

For an upper-tail probability,

.. math::

   P(X>x)=1-F(x).

For example,

.. math::

   P(X>6)=1-F(6)=0.40.


2.3.10 Relationship Between PDF and CDF
---------------------------------------

The PDF and CDF describe the same continuous distribution in different ways.

The CDF accumulates the PDF:

.. math::

   F(x)=\int_{-\infty}^{x}f(t)\,dt.

When :math:`F` is differentiable, the PDF can be recovered from the slope of
the CDF:

.. math::

   f(x)=F'(x).

For the waiting-time example, between 0 and 10,

.. math::

   F(x)=\frac{x}{10},

so

.. math::

   F'(x)=\frac{1}{10}=f(x).

Thus:

    PDF -> density and interval area

    CDF -> accumulated probability


2.3.11 Properties of a Continuous CDF
-------------------------------------

A CDF satisfies

.. math::

   0\leq F(x)\leq1.

It never decreases.

Also,

.. math::

   F(x)\rightarrow0
   \quad\text{as}\quad
   x\rightarrow-\infty,

and

.. math::

   F(x)\rightarrow1
   \quad\text{as}\quad
   x\rightarrow\infty.

For a continuous distribution with a continuous density, the CDF changes
smoothly rather than jumping at isolated values.


2.3.12 Discrete Versus Continuous Probability
---------------------------------------------

.. list-table::
   :header-rows: 1
   :widths: 30 35 35

   * - Idea
     - Discrete
     - Continuous
   * - Possible values
     - Countable
     - Any value in an interval
   * - Main probability function
     - PMF :math:`p(x)`
     - PDF :math:`f(x)`
   * - Probability at one value
     - May be positive
     - 0
   * - Probability over several values
     - Add PMF values
     - Find area under PDF
   * - CDF
     - Accumulates PMF values
     - Accumulates PDF area
   * - Typical CDF shape
     - Step function
     - Continuous curve


2.3.13 A Simple Method
----------------------

For a continuous probability distribution:

1. Identify the possible range of :math:`X`.
2. Identify the PDF :math:`f(x)`.
3. Check that :math:`f(x)\geq0` and the total area is 1.
4. Find probabilities as areas under the PDF.
5. Obtain the CDF by accumulating area from the left.
6. Use :math:`F(b)-F(a)` for interval probabilities when convenient.
7. Remember that individual points have probability 0.


2.3.14 Common Mistakes
----------------------

Do not write :math:`f(x)=P(X=x)` for a continuous random variable.

Do not interpret :math:`P(X=x)=0` as meaning that :math:`x` cannot occur.

Do not use density height alone as probability; probability is area.

Do not worry about whether interval endpoints are included.

Do not confuse the PDF :math:`f(x)` with the CDF :math:`F(x)`.


2.3.15 Summary
--------------

A continuous probability distribution is described by a PDF :math:`f(x)`.
A valid PDF satisfies

.. math::

   f(x)\geq0

and

.. math::

   \int_{-\infty}^{\infty}f(x)\,dx=1.

Probability is area:

.. math::

   P(a<X<b)=\int_a^b f(x)\,dx.

For any individual point,

.. math::

   P(X=x)=0.

The CDF is

.. math::

   F(x)=P(X\leq x)
   =
   \int_{-\infty}^{x}f(t)\,dt.

When differentiable,

.. math::

   f(x)=F'(x).


.. 2.3.16 Short Review Questions
.. -----------------------------

.. 1. Why does a continuous random variable use a PDF rather than a PMF?
.. 2. Why is :math:`P(X=x)=0` for an individual value?
.. 3. What conditions must a valid PDF satisfy?
.. 4. For the waiting-time example, find :math:`P(2<X<6)`.
.. 5. For the same example, find :math:`F(6)`.
.. 6. How are the PDF and CDF related?