Probability Engine

Mathematical Principles & Theorems

Studies probability spaces \((\Omega, \mathcal{F}, P)\) and random variable distributions: Discrete (Binomial \(P(k)=\binom{n}{k}p^k(1-p)^{n-k}\), Poisson \(P(k)=\frac{\lambda^k e^{-\lambda}}{k!}\)) and Continuous (Gaussian Normal \(\mathcal{N}(\mu, \sigma^2): f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\), Exponential \(f(x)=\lambda e^{-\lambda x}\)). Expectation \(\mathbb{E}[X] = \int x f(x)dx\), Variance \(\text{Var}(X) = \mathbb{E}[X^2] - (\mathbb{E}[X])^2\). Cumulative Distribution Function \(F(x) = P(X \le x)\).

Operating Instructions

• Select distribution: Normal, Poisson, Binomial, Exponential, or Uniform.
• Adjust parameters (\(\mu, \sigma, n, p, \lambda\)) using the precise numeric value boxes.
• Inspect live Probability Density Function (PDF) and Cumulative Distribution Function (CDF) curves.
• Compute exact interval probabilities \(P(a \le X \le b)\) and view shaded integral areas.