4.1 Recognizing Continuous Probability Models

Key Terms

Continuous random variable（連續型隨機變數）

A random variable that can take any value in an interval of real numbers.

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.

Cumulative distribution function（累積分配函數）

A function that gives \(P(X\leq x)\).

Parameter（參數）

A numerical value that controls the location, spread, or shape of a probability distribution.

Symmetric distribution（對稱分配）

A distribution whose left and right sides have the same shape around its center.

Right-skewed distribution（右偏分配）

A distribution with a longer tail toward larger values.

Rate（發生率）

The average number of events per unit time, distance, area, or volume.

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 \(X\), probabilities are represented by areas under a probability density function.

If \(f(x)\) is the probability density function of \(X\), then

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

A valid probability density function satisfies

\[f(x)\geq0\]

and

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

The total area under the density curve is 1.

For a continuous random variable,

\[P(X=x)=0.\]

Therefore,

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

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 \(X\) 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

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 \(X\);

  • the physical or experimental process;

  • the assumptions of the distribution;

  • the observed shape when data are available.