![[Pasted image 20240618025035.png|400]] # Distributions probability density functions aid in the analysis of continuous random variables, for which the probability of sampling a value in a specific range is the integration of the function over the interval Generally speaking, a *density* in the field of statistics is just a statistical measure that describes the likelihood of an outcome occurring. discrete states ## Terminology **Parametric methods** assume a specific distribution and estimate parameters while **non-parametric methods** rely on rankings or resampling techniques for analysis. posterior: what you think the parameters are post using data prior: what you think the parameters are prior to using data --- **Parametric methods** assume a specific distribution and estimate parameters while **non-parametric methods** rely on rankings or resampling techniques for analysis. ## Probability Distribution Functions (PDFs) > See also: > - [[Random Variables]] ### Discrete - Probability Mass Functions (PMFs) The **Probability Mass Function (PMF)** is a function that gives the probability that a discrete random variable is exactly equal to some value --- A **Probability Density Function (PDF)** of a continuous probability distribution is a function whose value at *any given sample/point* in the [[Probability Theory#The Sample Space|sample space]] can be considered the *relative likelihood* that the value of the random variable would be equal to that point. Can be used to specify the probability of a 1. The probability that $x$ is between two points $a$ and $b$ is $P(x)$ $P[a \le x \le b]$ 2. ### Cumulative Distribution Functions (CDFs) > [!summary] **Definition:** Cumulative Distribution Functions (CDF) > Contents ## Complex Probability Distributions ### Joint Probability Distributions ### Conditional Probability Distributions ### Convolution of Distributions > See also: > - [[Bayesian Statistics]] > - [[Mathematical Convolution]] https://en.wikipedia.org/wiki/Convolution_of_probability_distributions When you multiply the probability density functions (PDFs) of two independent random variables, you are performing a convolution This results in a PDF of the sum of these variables. \color{blue} ## Types of Distributions - [[Uniform Distributions]] - [[Gaussian Distributions]] - [[Binomial Distributions]] - [[The Beta Function]] - [[Dirichlet Distributions]] - [ ] Binomial - [ ] Poisson - [ ] Exponential - [ ] Skewed - [ ] Bimodal - [ ] Log-normal https://en.wikipedia.org/wiki/List_of_probability_distributions ![[Pasted image 20240612062809.png|350]]