Probability Distributions

Probability distributions are mathematical functions that describe the likelihood of different outcomes in a random process. They are fundamental tools in statistics, data science, and risk analysis. The Normal (Gaussian) distribution is the most widely used, describing many natural phenomena through its bell-shaped curve. The Exponential distribution models time between events in processes like radioactive decay or customer arrivals. The Uniform distribution represents equal probability across a range, useful for random number generation. The Poisson distribution counts discrete events occurring in fixed intervals, such as emails received per hour. Each distribution is characterized by specific parameters (like mean and standard deviation for Normal) that control its shape and location. Understanding these distributions enables hypothesis testing, confidence interval construction, and predictive modeling.

Start by selecting a distribution family from the dropdown menu - Normal, Exponential, Uniform, or Poisson. Enter the required parameters for your chosen distribution. For the Normal distribution, specify the mean (center) and standard deviation (spread). For Exponential, provide the rate parameter (λ). The Overview tab displays key statistics like mean, variance, and quantiles, along with four interactive charts: PDF (probability density), CDF (cumulative probability), survival function, and quantile function. Use the Tools tab for specific calculations: find P(X ≤ x) for cumulative probability, calculate P(a < X < b) for range probabilities, or determine the quantile for a given probability using inverse CDF. The Samples tab lets you generate random data from your distribution with a specified sample size and seed for reproducibility. Interactive visualizations update in real-time as you adjust parameters, with shaded regions highlighting probability areas under the curve.

Distribution statistics provide comprehensive insights into your probability model. The mean represents the central tendency or expected value. Variance and standard deviation measure the spread or dispersion of values around the mean. Quantiles (percentiles) show threshold values at specific probabilities - for example, the 95th percentile (p95) is the value below which 95% of observations fall. Skewness indicates asymmetry: positive skewness means the tail extends to the right, negative to the left. Kurtosis measures tail heaviness compared to the Normal distribution. When calculating probabilities, P(X ≤ x) gives the cumulative probability up to value x, while P(a < X < b) shows the probability of falling within a range. The inverse CDF (quantile function) works backwards - given a probability p, it finds the value x where P(X ≤ x) = p. The survival function S(x) = 1 - F(x) shows the probability of exceeding a value, commonly used in reliability analysis and survival studies.

Probability distributions power critical statistical applications across disciplines. In hypothesis testing, the Normal distribution determines z-scores and p-values for determining statistical significance of experimental results. The t-distribution (similar to Normal) handles small sample tests. Quality control uses Normal distributions to set control limits and detect manufacturing defects. Risk analysis employs distributions to model financial returns, insurance claims, and project completion times. The Exponential distribution is essential in reliability engineering for predicting equipment failures and maintenance schedules. Queuing theory uses Poisson and Exponential distributions to optimize service systems like call centers and traffic flow. Machine learning relies on distributions for probabilistic models, Bayesian inference, and generative algorithms. Weather forecasting, epidemiology, and seismology all use specialized distributions to model natural phenomena. Understanding which distribution fits your data enables accurate probability calculations, confidence intervals, and predictive modeling for data-driven decision making.