---
title: "probability distribution"
type: concept
related: [Gaussian Distribution, Probability Distributions, Normal Distribution, Baysian Network, Probability Mass Function]
source: https://www.jemoka.com/posts/kbhprobability_distributions/
confidence: high
status: active
---

probability distributions &ldquo;assigns probability to outcomes&rdquo;
\(X\) follows distribution \(D\). \(X\) is a &ldquo;\(D\) random variable&rdquo;, where \(D\) is some distribution (normal, gaussian, etc.)
syntax: \(X \sim D\).
Each distribution has three properties:
variables (what is being modeled) values (what values can they take on) parameters (how many degrees of freedom do we have) Types of Distribution discrete distribution described by PMF continuous distribution described by PDF parametrized distribution We often represent probability distribution using a set of parameters \(\theta_{j}\). For instance, a normal distribution is given by \(\mu\) and \(\sigma\), and a PMF is by the probability mass for each.
Methods of Compressing the Parameters of a Distribution So, for instance, for a binary distribution with \(n\) variables which we know nothing about, we have:
\begin{equation} 2^{n} - 1 \end{equation}
parameters (\(2^{n}\) different possibilities of combinations, and \(1\) non-free variables to ensure that the distribution add up)
assuming independence HOWEVER, if the variables were independent, this becomes much easier. Because the variables are independent, we can claim that:
\begin{equation} p(x_{1\dots n}) = \prod_{i}^{} p(x_{i)) \end{equation}
decision tree For instance, you can have a decision tree which you selectively ignore some combinations.
In this case, we ignored \(z\) if both \(x\) and \(y\) are \(0\).
Baysian networks see Baysian Network
types of probability distributions discrete distribution continuous distribution joint probability distribution distribution of note uniform distribution gaussian distributions Gaussian distribution Truncated Gaussian distribution uniform distribution \begin{equation} X \sim Uni(\alpha, \beta) \end{equation}
\begin{equation} f(x) = \begin{cases} \frac{1}{\beta -\alpha }, 0\leq x \leq 10 \\0 \end{cases} \end{equation}
\begin{equation} E[x] = \frac{1}{2}(\alpha +\beta) \end{equation}
\begin{equation} Var(X) = \frac{1}{12}(\beta -\alpha )^{2} \end{equation}
Gaussian Things Truncated Gaussian distribution Sometimes, we don&rsquo;t want to use a Gaussian distribution for values above or below a threshold (say if they are physically impossible). In those cases, we have some:
\begin{equation} X \sim N(\mu, \sigma^{2}, a, b) \end{equation}
bounded within the interval of \((a,b)\). The PDF of this function is given by:
\begin{equation} N(\mu, \sigma^{2}, a, b) = \frac{\frac{1}{\sigma} \phi \qty(\frac{x-\mu }{\sigma })}{\Phi \qty(\frac{b-\mu }{\sigma }) - \Phi \qty(\frac{a-\mu}{\sigma})} \end{equation}
where:
\begin{equation} \Phi = \int_{-\infty}^{x} \phi (x&rsquo;) \dd{x&rsquo;} \end{equation}
and where \(\phi\) is the standard normal density function.
three ways of analysis cumulative distribution function What is the probability that a random variable takes on value less tha
\begin{equation} cdf_{x}(x) = P(X&lt;x) = \int_{-\infty}^{x} p(x&rsquo;) dx' \end{equation}
sometimes written as:
\begin{equation} F(x) = P(X &lt; x) \end{equation}
Recall that, with
quantile function \begin{equation} \text{quantile}_{X}(\alpha) \end{equation}
is the value \(x\) such that:
\begin{equation} P(X \leq x) = \alpha \end{equation}
That is, the quantile function returns the minimum value of \(x\) at which point a certain cumulative distribution value desired is achieved.
adding uniform distribution for \(1 &lt; a &lt; 2\)
\begin{equation} f(X+Y = a) = \begin{cases} a, 0 &lt; a &lt; 1, \\ 2-a, 1 &lt; a &lt; 2, \\ 0, otherwise \end{cases} \end{equation}
