Other Common Distributions

Exponential Distribution

The exponential distribution is frequently used to model the time between independent events occurring at a constant average rate, such as the time until a radioactive particle decays or the time between bus arrivals at a stop. It's characterized by its continuous probability distribution and is described by a single parameter, lambda (λ), which is the rate parameter.

The probability density function (PDF) for the exponential distribution is given by:

$f(x|\lambda) = \lambda * e^{-\lambda x} \text{ for } x \geq 0$

Here, $e$ is the base of the natural logarithm, and $\lambda$ is the rate at which events occur. The exponential distribution is memoryless, meaning the probability of an event occurring in the future is independent of the elapsed time.

Exponential distribution probability density functions for different rate parameters (λ)

Uniform Distribution

The uniform distribution is one of the simplest probability distributions and represents a scenario where all outcomes are equally likely. It can be either discrete or continuous. In the continuous case, the uniform distribution is defined over an interval $[a, b]$, where every number in this interval has an equal probability of being drawn.

The PDF of a continuous uniform distribution is given by:

$f(x|a, b) = \frac{1}{b - a} \text{ for } a \leq x \leq b$

A common application of the uniform distribution is in simulations where random numbers are needed, and each number should have an equal chance of being selected.

Uniform distribution probability density functions for different intervals

Gamma Distribution

The gamma distribution is a two-parameter family of continuous probability distributions. It's often used to model waiting times or lifetimes of events, and it generalizes the exponential distribution. The two parameters are the shape parameter $(k)$ and the scale parameter $(\theta)$.

The gamma distribution's PDF is:

$f(x|k, \theta) = \frac{x^{k-1} * e^{-x/\theta}}{\theta^k * \Gamma(k)} \text{ for } x > 0$

where $\Gamma(k)$ is the gamma function, which extends the factorial function to continuous values. This distribution is particularly useful in Bayesian statistics and for modeling the distribution of sums of exponentially distributed random variables.

Gamma distribution probability density functions for different shape (k) and scale (θ) parameters

Beta Distribution

The beta distribution is a versatile distribution defined on the interval $[0, 1]$ and is parameterized by two positive shape parameters, $\alpha$ and $\beta$. It's often used in Bayesian inference and to model the behavior of random variables limited to a finite interval, such as proportions and probabilities.

The PDF of the beta distribution is:

$f(x|\alpha, \beta) = \frac{x^{\alpha-1} * (1-x)^{\beta-1}}{B(\alpha, \beta)} \text{ for } 0 \leq x \leq 1$

where $B(\alpha, \beta)$ is the beta function, a normalization constant that ensures the total probability integrates to 1. The beta distribution is particularly useful in machine learning for modeling prior distributions and for scenarios where outcomes are percentages or proportions.

Beta distribution probability density functions for different shape parameters (α, β)

Log-Normal Distribution

The log-normal distribution is applicable when modeling phenomena that result from the multiplicative effect of many independent random variables. If a random variable $X$ is log-normally distributed, then $Y = \ln(X)$ follows a normal distribution. This distribution is commonly used to model financial data, such as stock prices, and natural phenomena like the distribution of incomes.

The PDF of the log-normal distribution is:

$f(x|\mu, \sigma) = \frac{1}{x\sigma\sqrt{2\pi}} * e^{-(\ln(x) - \mu)^2 / (2\sigma^2)} \text{ for } x > 0$

where $\mu$ and $\sigma$ are the mean and standard deviation of the variable's natural logarithm. The log-normal distribution is positively skewed and is an effective model for data that cannot be negative and tend to cluster around a lower bound.

Log-normal distribution probability density functions for different scale (σ) parameters

Each of these distributions serves a unique purpose and provides tools to model different types of data. By understanding these distributions, you'll be better equipped to choose the right model for your data and make informed decisions in your machine learning projects. As you continue your journey into more advanced statistical techniques, keep these distributions in mind as essential building blocks for robust data analysis.