1. Discrete Distributions
Uniform Distribution (Discrete)
All outcomes have the same chance. Think of rolling a fair die (1 to 6).
PMF: P(X = x) = 1 / (b - a + 1), for x = a, a+1, ..., b
Support: {a, a+1, ..., b}
Mean: (a + b) / 2
Variance: ((b - a + 1)² - 1) / 12
Use case: Random number generation in games.
Bernoulli Distribution
A single trial with two outcomes: success (1) with probability p, and failure (0) with
probability (1 - p).
PMF: P(X = x) = px (1 - p)1 - x, for x ∈ {0,1}
Mean: p
Variance: p(1 - p)
Use case: Coin toss, yes/no experiments.
Binomial Distribution
Models the number of successes in n independent Bernoulli trials.
PMF: P(X = k) = C(n, k) × pk × (1 - p)n - k, where 0
≤ k ≤ n
Mean: np
Variance: np(1 - p)
Use case: Number of correct answers in a quiz, number of defective items in
a batch.
Poisson Distribution
Counts the number of events in a fixed interval of time or space, with known average rate λ.
PMF: P(X = k) = (λk e-λ) / k!, for k = 0, 1, 2, ...
Mean: λ
Variance: λ
Use case: Call center calls per hour, typing errors per page.
2. Continuous Distributions
Uniform Distribution (Continuous)
Every value in the interval [a, b] is equally likely.
PDF: f(x) = 1 / (b - a), for x in [a, b]
Mean: (a + b) / 2
Variance: (b - a)² / 12
Use case: Random number between two values (like random time delay).
Normal (Gaussian) Distribution
The most common distribution — shaped like a bell curve. Many natural and social phenomena
follow this.
PDF: f(x) = (1 / √(2πσ²)) × e-(x - μ)² / (2σ²)
Mean: μ
Variance: σ²
Use case: Heights, IQ scores, measurement errors.
Exponential Distribution
Models time until the next event. It's memoryless — past events don’t affect the future.
PDF: f(x) = λe-λx, for x ≥ 0
Mean: 1 / λ
Variance: 1 / λ²
Use case: Time between arrivals at a bus stop, battery life.
Chi-Squared Distribution
Arises from the sum of squares of k independent standard normal variables.
PDF: Involves gamma function, defined for x > 0
Mean: k
Variance: 2k
Use case: Hypothesis testing, goodness-of-fit tests.
t-Distribution (Student's t)
Used when the sample size is small and population variance is unknown. Similar to the normal
curve but with heavier tails.
PDF: Involves the gamma function and degrees of freedom (df)
Mean: 0 (if df > 1)
Variance: df / (df - 2), for df > 2
Use case: Confidence intervals and t-tests.
3. Summary
- Discrete Distributions: Used for countable outcomes like number of heads in 10 coin tosses.
- Continuous Distributions: Used for measurable outcomes like weight, time, or distance.
- PMF (Probability Mass Function): Used for discrete distributions to assign probability to exact values.
- PDF (Probability Density Function): Used for continuous distributions; area under the curve gives probability.
- Each distribution has specific parameters, along with a formula for mean and variance.
- Choosing the right distribution depends on the data type and the context of the problem.