4.1 Recognizing Continuous Probability Models
=============================================

:bolditalic:`Key Terms`

:bolditalic:`Continuous random variable（連續型隨機變數）`
   A random variable that can take any value in an interval of real numbers.

:bolditalic:`Probability density function（機率密度函數）`
   A function whose area over an interval gives the probability that the random variable falls in that interval.

:bolditalic:`Cumulative distribution function（累積分配函數）`
   A function that gives :math:`P(X\leq x)`.

:bolditalic:`Parameter（參數）`
   A numerical value that controls the location, spread, or shape of a probability distribution.

:bolditalic:`Symmetric distribution（對稱分配）`
   A distribution whose left and right sides have the same shape around its center.

:bolditalic:`Right-skewed distribution（右偏分配）`
   A distribution with a longer tail toward larger values.

:bolditalic:`Rate（發生率）`
   The average number of events per unit time, distance, area, or volume.

:bolditalic:`Lifetime distribution（壽命分配）`
   A probability model used for positive quantities such as time to failure or service life.

4.1.2 Learning Outcomes
-----------------------

After completing this chapter, students should be able to:

- explain how probabilities are represented for a continuous random variable;
- recognize the main structure of common continuous probability distributions;
- calculate probabilities for continuous uniform and normal random variables;
- standardize a normal random variable and use standard normal probabilities;
- use the normal distribution to approximate selected discrete probabilities;
- calculate waiting-time probabilities with the exponential distribution;
- explain the relationship among Poisson, exponential, gamma, and Erlang models;
- recognize when Weibull and lognormal distributions are useful for lifetime data;
- recognize when a beta distribution is appropriate for a continuous proportion;
- compare the support, shape, and main parameters of common continuous distributions;
- select a reasonable continuous probability model from the description of an experiment.

4.1.3 A Short Review: Probability Is Area
-----------------------------------------

For a continuous random variable :math:`X`, probabilities are represented by
areas under a probability density function.

If :math:`f(x)` is the probability density function of :math:`X`, then

.. math::

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

A valid probability density function satisfies

.. math::

   f(x)\geq0

and

.. math::

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

The total area under the density curve is 1.

For a continuous random variable,

.. math::

   P(X=x)=0.

Therefore,

.. math::

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

For continuous variables, including or excluding one endpoint does not change
the probability.

4.1.4 From a General Density to a Named Distribution
----------------------------------------------------

A probability density can have many different shapes.

Some shapes occur often enough that they have standard names and formulas.

For example:

- a flat density on a finite interval leads to the **continuous uniform**
  distribution;
- a symmetric bell-shaped density leads to the **normal** distribution;
- a right-skewed waiting-time density with a constant event rate leads to the
  **exponential** distribution;
- waiting until several events occur leads naturally to the **gamma** or
  **Erlang** distribution;
- flexible lifetime models often use the **Weibull** distribution;
- positive variables whose logarithms are normal use the **lognormal**
  distribution;
- continuous proportions between 0 and 1 can often be modeled with the
  **beta** distribution.

The main idea is

    understand what :math:`X` measures

    -> identify its possible values

    -> identify the shape or process

    -> choose a distribution

    -> identify the parameters

    -> calculate the required probability.

4.1.5 Preview of the Main Models
--------------------------------

.. list-table::
   :header-rows: 1
   :widths: 22 35 43

   * - Distribution
     - Typical Random Variable
     - Main Structure
   * - Continuous uniform
     - A value within a fixed interval
     - Equal density throughout the interval
   * - Normal
     - Measurement or error
     - Symmetric bell-shaped behavior
   * - Exponential
     - Waiting time to the next event
     - Constant event rate and memoryless waiting time
   * - Gamma / Erlang
     - Waiting time to several events
     - Sum of exponential waiting times
   * - Weibull
     - Lifetime or time to failure
     - Flexible failure behavior
   * - Lognormal
     - Positive lifetime, size, or rate
     - Natural logarithm is normally distributed
   * - Beta
     - Continuous proportion
     - Bounded between 0 and 1 with flexible shape

4.1.6 Common Mistake: Choosing by Shape Alone
---------------------------------------------

A graph can help identify a model, but shape alone is not enough.

For example, exponential, gamma, Weibull, and lognormal distributions can all
be right-skewed.

The process that produces the data also matters.

A good model should match:

- the possible values of :math:`X`;
- the physical or experimental process;
- the assumptions of the distribution;
- the observed shape when data are available.