Probability Distribution

Statements

Definition. Let $(\Omega ,\mathcal{F},P)$ be a probability space, $(\Gamma ,\mathcal{S})$ be a measurable space and $X:\Omega \to \Gamma$ be a random variable, then the function $P_X:\mathcal{S}\to [0,1]$ defined as $S\mapsto P(X^{-1})(S)$ is said to be the probability distribution of the random variable $X$.

Interpretation

Intuitively, the notion of randomness in the values $X(\omega )$ of $X$ is captured by this construction as follows:

1. An element $\omega \in \Omega$ is selected according to the probability $P(\{\omega \})$ that is allocated to $\omega$ by the probability measure $P$ on $(\Omega ,\mathcal{F})$.
2. This $\omega$ is mapped onto an element $X(\omega )$ in $\Gamma$, which is also referred to as a realization of the random variable $X$. Across realizations, the values of $X$ exhibit a probability distribution that depends both on the properties of $P$ and $X$ and is denoted by $P_X$. The Figure below visualizes the situation.

Clearly, if $\mathcal{X}=\Omega$, $\mathcal{S}=\mathcal{F}$ and $X:=$ id, then $P$ and $P_X$ are identical. Importantly, the union of the measurable space $(\mathcal{X},\mathcal{S})$ and the probability measure $P_X$ forms the probability space $(\mathcal{X},\mathcal{S},P_X)$.

Notation

We first note that random variables of the form $X:\Omega \to \mathcal{X}$ are often written as $X:(\Gamma ,\mathcal{F})\to (\mathcal{X},\mathcal{S})$. Second, the following notational conventions for events in $\mathcal{F}$ are commonly employed

$$\{X\in S\}=\{\omega \in \Omega |X(\omega )\in S\}$$ $$\{X\diamond x\}=\{\omega \in \Omega |X(\omega )\diamond x\},\diamond \in \{>,\ge ,=,\le ,<\}$$

for $S\in \mathcal{S}$ and $x\in \mathcal{X}$. The probability $P$ is the Measure of the above sets, e.g., $P(\xi \in S)=P(\{\omega \in \Omega |\xi (\omega )\in S\})$

Discrete Distribution

Probability Mass Function

Probability Mass Function

Probability mass functions are used to define the distributions of discrete random variables $X:\Omega \to \mathcal{X}$ with discrete and finite (or least countable) outcome set $\mathcal{X}$.

Statements

Definition. Let $(\Omega ,\mathcal{F},P)$ be a probability space. A random variable $X:\Omega \to \mathcal{X}$ is said to be discrete, if its outcome space $\mathcal{X}$ contains countably many elements $x_i,i=1,2,\dots$. The probability mass function (PMF) of a discrete random variable $X$ is denoted by $p_X$ and is defined as

$$p_X:\mathcal{X}\to [0,1]$$ $$p_X(x_i)=P_X(X=x_i)$$

References

D. Ostwald - The general linear model 20_21

Probability mass function - Wikipedia

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Continuous Distribution

Probability Density Function

Probability Density Function

Probability density functions are used to define the distributions of continuous random variables $X:\Omega \to \mathbb{R}$. We use the following definitions.

Statements

Definition. The probability density function (PDF) of a continuous Random Variable is defined as a function $p_X:\mathbb{R}\to\mathbb{R}_{\geq0}$$ with the properties

1. ${\int }_{\mathbb{R}}{p}_X(x)dx=1$
2. $p_X(x_1\le x\le x_2)={\int }_{x_1}^{x_2}{p}_X(x)dx,\forall x_1,x_2\in \mathbb{R}$ with $x_1\le x_2$.

References

D. Ostwald - The general linear model 20_21

Probability density function - Wikipedia

Link to original

Most Common Distributions

TABLE title AS "Title",
	category AS "Category",
	is_continuous AS "Is_Continuous"
WHERE contains(title, "Distribution") AND title != "Probability Distribution"
SORT title ASC, category ASC

References

D. Ostwald - The general linear model 20_21

Probability distribution - Wikipedia