Probability Distributions

Probability distributions model the likelihood of outcomes for random variables, forming the backbone of statistics and data science. At MathMultiverse, we simplify these concepts with clear explanations, interactive visualizations, and practical examples, from normal distributions in natural phenomena to binomial outcomes in trials.

These tools drive decision-making in finance, medicine, and more, helping quantify uncertainty and predict patterns.

Normal Distribution

The normal distribution, with its bell-shaped curve, is defined by mean (\( \mu \)) and standard deviation (\( \sigma \)).

\[ f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x - \mu)^2}{2\sigma^2}} \]

Properties: Symmetric, 68% within 1\( \sigma \), 95% within 2\( \sigma \).

Z-Score

\[ z = \frac{x - \mu}{\sigma} \]

Example: Exam Scores

\( \mu = 75 \), \( \sigma = 8 \). Find \( P(X > 85) \):

\[ z = \frac{85 - 75}{8} = 1.25 \] \[ P(Z > 1.25) \approx 0.1056 \]

Normal Distribution

Binomial Distribution

Models successes in \( n \) trials with probability \( p \).

\[ P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} \]

Mean: \( \mu = np \), Variance: \( \sigma^2 = np(1 - p) \).

Example: Coin Flips

\( n = 10 \), \( p = 0.5 \). Find \( P(X = 6) \):

\[ P(X = 6) = \binom{10}{6} (0.5)^6 (0.5)^4 \approx 0.2051 \]

Poisson Approximation

For \( n = 100 \), \( p = 0.01 \), \( \lambda = 1 \). Find \( P(X = 2) \):

\[ P(X = 2) = \frac{1^2 \cdot e^{-1}}{2!} \approx 0.1839 \]

Binomial Distribution

Examples

Normal: Exam Scores

\( \mu = 75 \), \( \sigma = 8 \). Find \( P(X > 85) \):

\[ z = \frac{85 - 75}{8} = 1.25 \] \[ P(Z > 1.25) \approx 0.1056 \]

Binomial: Survey

\( n = 50 \), \( p = 0.4 \). Find \( P(X = 20) \):

\[ P(X = 20) \approx 0.1123 \]

Poisson: Arrivals

\( \lambda = 5 \). Find \( P(X = 3) \):

\[ P(X = 3) = \frac{5^3 \cdot e^{-5}}{3!} \approx 0.1404 \]

Applications

  • Finance: Stock returns, \( P(\text{Return} > 1.5\%) \approx 0.0808 \).
  • Quality Control: Defects, \( P(X \leq 3) \approx 0.8648 \).
  • Medicine: Disease cases, \( P(X \geq 4) \approx 0.1429 \).
  • Meteorology: Temperature, \( P(X > 33) \approx 0.0475 \).
  • Marketing: Clicks, \( P(X = 10) \approx 0.1251 \).