Interactive PDF, CDF, and parameter exploration for common distributions
A probability distribution is a map of how likely each outcome is. Reshape them, see how they connect, and sample from them until the true shape appears.
Choose a distribution and drag its parameter sliders to shift, stretch, and reshape the PDF. Toggle the CDF overlay to see probabilities pile up from 0 to 1.
The bell curve -- most important distribution in statistics
A probability distribution describes how likely different outcomes are. The probability density function (PDF) gives the relative likelihood of each value, while the cumulative distribution function (CDF) gives the probability of being at or below a value. Every distribution is defined by a few parameters that control its shape, center, and spread.
Choose a distribution and drag the parameter sliders to see how the PDF changes in real time. Toggle the CDF overlay to see cumulative probabilities. Notice how changing parameters shifts, stretches, or reshapes the distribution.
Key insight: The total area under any PDF is exactly 1 -- it represents 100% of all possible outcomes. The CDF always starts at 0 and ends at 1, increasing monotonically. At any point x, CDF(x) = P(X ≤ x).
Hover over each node to see how one distribution grows out of another, from Bernoulli to Binomial to Normal.
Hover over distributions to highlight their connections. Each node shows a mini PDF. Arrows indicate how one distribution arises from or converges to another.
Distributions don't exist in isolation -- they form a rich web of connections. The Binomial arises from summing Bernoulli trials. As n grows, the Binomial converges to the Normal (Central Limit Theorem). The Gamma generalizes the Exponential. Hover over each node to explore these relationships.
Key insight: Understanding how distributions relate to each other helps you choose the right model. For instance, if you're counting rare events, use Poisson; if modeling wait times, use Exponential; if summing many small effects, the result is approximately Normal.
Pick a distribution and draw samples. Try just 10 from the bimodal, then 10,000, and watch the histogram settle onto the smooth curve.
Draw samples from any distribution and watch the histogram build up. The dashed line shows the sample mean. Compare the empirical shape to the theoretical PDF.
Draw random samples from any distribution and watch the empirical histogram take shape. With few samples, the histogram is noisy and irregular. As you add more samples, it converges to the smooth theoretical PDF -- a visual proof of the law of large numbers at work.
Key insight: Try drawing just 10 samples from the bimodal distribution -- you might not even see two modes. With 10,000 samples, the two peaks become unmistakable. This illustrates why sample size matters for statistical inference.
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