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  5. Continuous Probability Distributions

Continuous Probability Distributions

A continuous probability distribution describes the probabilities of a continuous random variable, which can take any fractional or decimal value within a given range. Unlike discrete distributions such as the binomial or hypergeometric distributions which deal with countable outcomes, continuous distributions are represented by smooth probability density curves.

Some key characteristics of continuous probability distributions include:

  • Probability Density Function (PDF): Instead of specific values, probabilities are found by measuring the area Under the curve within a certain range
  • Skewness: The shape of the distribution can be symmetrical, positively skewed (stretched to the right), or negatively skewed (stretched to the left)
  • Modality: A distribution can have one peak (unimodal), or two peaks, (bimodal), which may happen when there are two different groups in the data

Some examples of Common Continuous Distributions include:

  • Normal Distribution: This looks like a bell curve and is common in nature (i.e. heights, test scores)
  • Exponential Distribution: This is used for things like waiting time (i.e. time between bus arrivals)
  • Uniform Distribution: All values in a uniform distribution have equal chances (i.e. random number generation within a range)

Exponential Distribution

Exponential distribution helps predict how long you will wait between random events, like phone calls, customer arrivals, or machine failures.

Exponential Distribution can be expressed algebraically as:

\(P(x) = k e^{-kx}\)

Where:

  • \(k\) is the Rate Parameter, \(\cfrac{1}{\mu}\)
  • \(x\) is the Waiting Time
  • \(e\) is a Mathematical Constant, approximately \(2.71828\)

The Difference Between Exponential and Normal Distribution

The exponential distribution is asymmetric, meaning it has a peak at zero and a long tail on one side. This means events are most likely to happen quickly (i.e. a short wait), and the chance of longer waits decreases. It’s used for situations like waiting times or time between random events.

The normal distribution (or bell curve) is symmetric, meaning both sides of the curve are the same. Most values are close to the middle (Mean), and fewer values are found at the extremes. It’s often used to model things like people’s heights, where most are near average and fewer are very short or tall.


Example

Give three examples for each of these distributions: exponential, bimodal, and positively skewed distribution.

The following examples illustrate real-world scenarios for each distribution type. These cases highlight how the shape and properties of each distribution apply to different situations.

Bimodal Distribution

  • Test Scores: Most students score very high or low, with few in the middle.
  • Heights of Men and Women: Two peaks for men’s and women’s average heights.
  • Job Salaries: Different salary ranges for junior and senior-level employees.

Exponential Distribution

  • Time Between Bus Arrivals: Short waits are common, long waits are rare.
  • Radioactive Decay: Particles are more likely to decay quickly.
  • Customer Arrival Time at a Store: Customers arrive randomly, with short waits more likely.

Positively Skewed Distribution

  • Income Distribution: Most people earn average or low incomes, with a few high earners.
  • Age at Retirement: Most retire at a typical age, but some much later.
  • Number of Children in Families: Most families have 1-3 children, with a few having many more.

Suppose an Average Time of a customer at a service desk is 4 minutes. Find the probability density that the next customer arrives exactly at 2 minutes after the last arrival.

First, we can identify the following values:

  • \(\mu = 4\)
  • \(x = 2\)

Next, we can find \(k\) as such:

\(k = \cfrac{1}{\mu}\)

\(k = \cfrac{1}{4}\)

\(k = 0.25\)

Then, we can plug the appropriate values into the Exponential Distribution formula to determine the probablity density:

\(P(x) = ke^{-kx}\)

\(P(2) = 0.25 e^{-0.25 \cdot 2}\)

\(P(2) = 0.25 e^{-0.5}\)

\(P(2) = 0.25 \times 0.6065\)

\(P(2) \approx 0.1516\)

Therefore, we can determine that the Probability Density at \(x = 2\) minutes is approximately 0.1516.