Hassan Ijaz
Ai, Web & Design
Probability distributions
Interactive distribution playground with sliders for parameters, showing how shapes change and allowing overlay comparisons between different distributions
Concept Overview
Probability distributions describe how the probabilities of different outcomes are spread across the possible values of a random variable. They are the foundation for statistical inference and modeling.
Common Discrete Distributions
Binomial Distribution
- Models number of successes in n independent trials
- Parameters: n (trials), p (success probability)
- Example: Number of heads in 10 coin flips
- Mean: μ = np, Variance: σ² = np(1-p)
Poisson Distribution
- Models count of events in fixed time/space
- Parameter: λ (average rate)
- Example: Number of emails per hour
- Mean = Variance = λ
Geometric Distribution
- Models trials until first success
- Parameter: p (success probability)
- Example: Rolls until first 6 on a die
- Mean: 1/p, Variance: (1-p)/p²
Common Continuous Distributions
Normal (Gaussian) Distribution
- Bell-shaped, symmetric distribution
- Parameters: μ (mean), σ² (variance)
- Central Limit Theorem: sums tend toward normal
- 68-95-99.7 rule for standard deviations
Exponential Distribution
- Models time between events
- Parameter: λ (rate)
- Memoryless property
- Mean: 1/λ, Variance: 1/λ²
Uniform Distribution
- Equal probability across interval [a,b]
- Parameters: a (min), b (max)
- Mean: (a+b)/2
- Variance: (b-a)²/12
Key Concepts
- Parameters: Values that define the shape and location of a distribution
- Support: The set of possible values (e.g., { 0,1,2,... } for Poisson)
- Moments: Mean (1st), Variance (2nd), Skewness (3rd), Kurtosis (4th)
- Transformations: How distributions change under operations (e.g., sum of normals is normal)
Pro tip: Many real-world phenomena follow these distributions! Heights are normal, waiting times are exponential, and counts of rare events are Poisson. Understanding which distribution fits your data is crucial for proper statistical analysis.
Use the interactive playground below to adjust distribution parameters and see how shapes change. Try overlaying different distributions to compare their properties!
Interactive Visualization
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Interactive distribution playground with sliders for parameters, showing how shapes change and allowing overlay comparisons between different distributions