Key Concepts of Probability Density Functions to Know for Intro to Probability

Probability Density Functions (PDFs) describe how probabilities are distributed across different outcomes. Understanding various distributions, like uniform, normal, and exponential, is crucial in engineering and statistics for modeling real-world phenomena and making informed decisions.

1. Uniform distribution

   • Represents a constant probability across a defined interval, meaning every outcome is equally likely.
   • Defined by two parameters: the minimum (a) and maximum (b) values of the interval.
   • The probability density function (PDF) is flat, indicating no preference for any value within the range.

2. Normal (Gaussian) distribution

   • Characterized by its bell-shaped curve, symmetric around the mean (μ).
   • Defined by two parameters: mean (μ) and standard deviation (σ), which determine the center and spread of the distribution.
   • Many natural phenomena and measurement errors follow this distribution, making it fundamental in statistics.

3. Exponential distribution

   • Models the time until an event occurs, such as failure rates in reliability engineering.
   • Defined by a single parameter, the rate (λ), which is the inverse of the mean.
   • The PDF decreases exponentially, indicating that the likelihood of an event decreases over time.

4. Gamma distribution

   • A generalization of the exponential distribution, useful for modeling waiting times for multiple events.
   • Defined by two parameters: shape (k) and scale (θ), which influence the form of the distribution.
   • The PDF can take various shapes, including exponential and chi-square, depending on the parameter values.

5. Beta distribution

   • Defined on the interval [0, 1], making it suitable for modeling proportions and probabilities.
   • Characterized by two shape parameters (α and β) that determine the distribution's shape.
   • Flexible in form, allowing for various shapes such as uniform, U-shaped, or J-shaped distributions.

6. Chi-square distribution

   • Primarily used in hypothesis testing and confidence interval estimation for variance.
   • Defined by degrees of freedom (k), which affects the shape of the distribution.
   • The PDF is skewed to the right, especially for small degrees of freedom, and approaches normality as k increases.

7. Student's t-distribution

   • Used for estimating population parameters when sample sizes are small and population variance is unknown.
   • Defined by degrees of freedom (ν), which influence the distribution's tails and peak.
   • The PDF is similar to the normal distribution but has heavier tails, providing a more accurate estimate for small samples.

8. F-distribution

   • Used primarily in analysis of variance (ANOVA) and comparing variances between two populations.
   • Defined by two sets of degrees of freedom (d1 and d2) corresponding to the numerator and denominator.
   • The PDF is right-skewed, and the distribution approaches normality as the degrees of freedom increase.

9. Weibull distribution

   • Commonly used in reliability analysis and life data modeling.
   • Defined by two parameters: shape (k) and scale (λ), which determine the failure rate behavior.
   • The PDF can model increasing, constant, or decreasing failure rates depending on the shape parameter.

10. Lognormal distribution

    • Models variables that are positively skewed and cannot take negative values, such as income or stock prices.
    • A variable is log-normally distributed if its logarithm is normally distributed.
    • Defined by two parameters: mean (μ) and standard deviation (σ) of the logarithm of the variable, influencing the shape of the distribution.