l4- Introduction to Continuous Probability Distributions in Biology

Course Resources and Success Strategies

• Academic Skills Center Resources: Students are encouraged to use the numeracy and statistics modules provided by the Academic Skills Center via Learn.

  • These modules are particularly helpful for foundational instruction if topics in the Core Essentials Test felt unfamiliar or unclear.

  • Links to these resources are found in test feedback boxes and the BIO209 resources page.

• Instructional Materials:

  • Lecture slides, detailed notes, and code are provided weekly.

  • Note: For the previous week, no code or notes were generated, but they resume this week.

• Echo recordings:

  • Lectures and labs from weeks one and two are recorded.

  • Lab recordings will cease after week two as the format shifts from guided coding to group-based and self-directed learning.

• Textbook and Practice: Weekly practice problems and textbook-associated resources are available on the Learn page.

• The Learning Community:

  • Classmates are considered one of the most vital resources for troubleshooting and understanding practice problems.

  • Starting in week three labs, students will be assigned to a specific group to facilitate collaborative learning.

• Class Representation: The class representative is Erin Kang. Students can reach out to her for anonymous feedback to the teaching team or for help connecting with fellow students.

Introduction to Continuous Distributions

• Learning Objectives:

  • Describe the three most common continuous distributions (Uniform, Gaussian, and Log-normal) and their properties.

  • Choose the appropriate null distribution for biological observations.

  • Identify and interpret the Probability Density Function (PDF) and Cumulative Distribution Function (CDF).

• Conceptual Shift: Unlike discrete distributions, which deal with counts or categorized outcomes, continuous distributions focus on numeric values where any value within a range is possible.

• Continuous Uniform Distribution:

  • Case Study: Antarctic Ice Fish: Observed in 2022, ice fish nests in Antarctic waters show a highly regular, equally spaced pattern. The distance between these nests (measured in millimeters or centimeters) serves as an example of a continuous uniform distribution.

  • Definition: A distribution for numeric values between a specific minimum and maximum where every outcome within that range is equiprobable.

  • Functional Form: The distribution looks like a continuous line rather than discrete bars. Probability is calculated based on the area beneath the curve.

  • Cumulative Distribution Function (CDF): For a continuous uniform distribution, the CDF appears as a straight line with a steady, constant increase in probability as previous probabilities are summed.

  • Biological Null Hypothesis Application: It can model a null expectation where events occur with equal likelihood over a duration, such as the time intervals between a bumblebee's visits to flowers.

Gaussian (Normal) Distribution

• History and Terminology: Traditionally named after Carl Friedrich Gauss (Gaussian), it is interchangeably referred to as the Normal distribution.

• Case Study: Crown of Thorns Sea Stars ($Acanthaster\,planci$):

  • Researchers compared sea star diameters in The Maldives (outbreaks since 1970s) and the Great Barrier Reef (outbreaks since 1962).

  • Data was visualized using histograms, where observations were binned (e.g., $0\,cm$ to $10\,cm$, $10\,cm$ to $20\,cm$).

  • Both locations showed a central tendency: Langan Fushi Island stars peaked around $40\,cm$, while Rib Reef stars peaked between $25\,cm$ and $35\,cm$.

• Parameters:

  • Mean: Represented by the population parameter $\mu$.

  • Standard Deviation: Represented by the population parameter $\sigma$. Note that Variance is calculated as $\sigma^2$.

• Sample Space: Technically ranges from negative infinity ($-\infty$) to positive infinity ($\infty$).

• Properties:

  • The distribution is symmetric (a bell curve). If flipped at the mean, it looks identical.

  • Low variance results in a tall, narrow curve.

  • High variance results in a shorter, wider curve.

  • Shifting the mean ($\mu$) moves the center of the distribution along the x-axis.

• Limitations in Biology: While widely used to approximate biological systems, the Gaussian distribution assumes outcomes can be negative, which is often biologically impossible (e.g., a sea star cannot have a negative diameter).

Log-normal Distribution

• Definition: A continuous distribution where the natural logarithm of the variable is normally distributed.

• Sample Space: Limited from $0$ to positive infinity ($\infty$). This boundary makes it more biologically realistic for measurements that cannot be negative.

• Comparison to Gaussian:

  • Used when the variance is large and the distribution is centered close to zero.

  • Unlike the Gaussian distribution, the log-normal distribution is skewed and not symmetrical.

  • As variance increases, the peak of the distribution shifts away from the mean due to the skew.

• Examples of Log-normal Variables:

  • Daily Rainfall: Small amounts or zero rain are common, while extreme rain events are rare, and negative rainfall is impossible.

  • Distance Traveled: Movement by organisms in a specific context.

  • Solar Radiation: Measured at the leaf level; varies between zero (night) and high values (clear day).

  • Protein Expression: Some cells produce no protein, while others produce massive amounts.

Interlude: Understanding Logarithms

• Definition: The logarithm to base $b$ is the inverse of exponentiation with base $b$.

• Conceptual Example: If $10^3 = 1000$, then $\log_{10}(1000) = 3$.

• Natural Logarithm (ln): The log-normal distribution specifically utilizes the natural log, which has the mathematical constant $e$ as its base.

• The Constant $e$: Approximately equal to $2.71828$.

• Standard Usage: The natural log is the default logarithm used in R and throughout this biology course.

Selecting Appropriate Distributions for Biological Data

• Decision Matrix (Class Exercise):

  • Clutch Size (Albatross Pairs): Typically a Poisson distribution. Because it is a count of offspring (discrete), a continuous distribution like log-normal is inappropriate, even though both exclude negative values.

  • Time Students Spend in Class: Ideally a Log-normal (or Normal). It cannot be negative and usually peaks around a central value determined by course credit hours.

  • Proportion of Classes Attended: Best modeled by a Binomial distribution. While a proportion looks continuous (between $0$ and $1$), it is actually an accumulation of binary Bernoulli trials (attended vs. not attended).

  • Number of Zooplankton in Seawater: Typically Poisson or Discrete Uniform. Because it is a "number," it requires a discrete distribution. If measuring the density of zooplankton per liter, a continuous distribution might be used instead.

  • Heights of Students: Gaussian (Normal). In adult populations, height tends to be symmetric around a mean far from zero, making the negative-value limitation of Gaussian distributions irrelevant.

Questions & Discussion

• Student Question: Does $\sigma$ mean standard deviation or variance?

• Instructor Response: $\sigma$ refers to the standard deviation. The variance is the square of the standard deviation ($\sigma^2$). In the formula examples, different curves were generated by adjusting these variance/standard deviation settings.

• Student Question: Regarding student class time, could that be an upper-bound issue for Gaussian distributions?

• Instructor Response: Yes, there is a theoretical upper bound (total hours in a week). However, in Gaussian and Log-normal models, values extremely far from the mean have such infinitesimally small probabilities that they are often accepted as functional assumptions for the null distribution.