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Unit A: Biostatistics

7. Probability Distributions
Categorizing and modeling the behavior of discrete and continuous biological data.
Probability Distributions
Discrete Distributions
(Countable data)

• Binomial Distribution

• Poisson Distribution

Continuous Distributions
(Measurable data)

• Normal Distribution

A. Binomial Distribution
Modeling "Success" vs "Failure" in fixed trials.

Models the probability of a specific number of "successes" in a fixed number of independent trials, where each trial has only two possible outcomes.

Key Conditions:

  • The number of trials (
    n
    ) is fixed in advance.
  • The probability of success (
    p
    ) remains constant.
  • The trials are independent (Probability of failure
    q = 1 - p
    ).

Probability Mass Function (PMF):

P(X = k) = ⁿCₖ · pᵏ · qⁿ⁻ᵏ

Where ⁿCₖ represents the number of ways to arrange those successes.

Mean (μ)

np

Variance (σ²)

npq

S.D. (σ)

√npq

B. Poisson Distribution
Modeling rare events in fixed intervals.

Models the number of times a rare event occurs within a fixed interval of time or space.

Crucial Identifier: The probability of a single success (

p
) is extremely small, but the number of trials (
n
) is extremely large.

Biological Examples:

  • Number of rare genetic mutations occurring per kilobase of DNA.
  • Distribution of parasites on a host fish.
  • Number of cellular colonies growing on an agar plate.

Probability Mass Function (PMF):

P(X = k) = (λᵏ · e⁻λ) / k!

Where λ is the average number of occurrences, and e ≈ 2.718.

Unique Mathematical Property

Mean (μ) = Variance (σ²) = λ

C. Normal (Gaussian) Distribution
The cornerstone of continuous biostatistics.

Maps variables influenced by a massive number of tiny, independent, random genetic and environmental factors (e.g., human height, enzyme levels, egg weight).

Key Properties:

  • Forms a perfectly symmetrical, bell-shaped curve.
  • The Mean, Median, and Mode are all perfectly equal and sit at the center.
  • The total area under the curve is exactly 1.0 (100% probability).
  • Defined by just two parameters: Mean (μ) and Standard Deviation (σ).

Standard Normal Distribution (Z-score)

To compare different normal distributions, we standardize them into Z-scores:

Z = (X - μ) / σ

A Standard Normal Curve always has:
Mean = 0 | S.D. = 1

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