Probability Distributions

4. Probability Distributions: Random Variables

Probability Distributions

Probability Distribution

Definition

A probability distribution is a function that links each outcome $\,$ of a random experiment to the probability of its occurrence.

Notation

The probability distribution of a random variable $X$ is denoted by $f(x)$ .

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Discrete Variables: Probability Mass Function

The probability distribution of a discrete random variable is called the probability mass function and gives the probability that $X$ equals $x$ , for each $x$ within the range of $X$ .

$f(x)=\mathbb{P}(X=x)$

The cumulative distribution function of a discrete random variable gives the probability that $X$ is less than or equal to $x$ .

$F(x)=\mathbb{P}(X\leq x)$

Consider the random experiment of tossing two coins. Let $X$ denote the number of Heads.

Then the probability distribution of $X$ is:

$x$	$0$	$1$	$2$
$\mathbb{P}(X=x)$	$0.25$	$0.5$	$0.25$

Discrete Distribution: Important Properties

For any discrete probability distribution, the following properties hold:

$\mathbb{P}(X \leq x) = \mathbb{P}(X=0) +\mathbb{P}(X=1) + \ldots + \mathbb{P}(X=x)$
$\mathbb{P}(X \geq x) = 1 - \mathbb{P}(X \lt x) = 1 - \mathbb{P}(X \leq x-1)$
$\mathbb{P}(X \lt x) = \mathbb{P}(X \leq x-1) = 1 - \mathbb{P}(X \geq x)$
$\mathbb{P}(X \gt x) = \mathbb{P}(X \geq x+1) = 1 - \mathbb{P}(X \leq x)$

Additionally, for any $2$ values $a$ and $b$ in the range of $X$ (with $a<b$ ):

$\mathbb{P}(a \lt X \leq b) = \mathbb{P}(X \leq b) - \mathbb{P}(X \leq a)$

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Continuous Variables: Probability Density Function

The probability distribution of a continuous random variable is called the probability density function.

The probability that a random variable $X$ lies between $a$ and $b$ is given by the area under the curve
from $a$ to $b$ .
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Continuous Distribution: Important Properties

For any continuous probability distribution, the following properties always hold:

$\mathbb{P}(X=k)=0$
$\mathbb{P}(X\leq k)=\mathbb{P}(X\lt k) =1 - \mathbb{P}(X \geq k)$
$\mathbb{P}(X\geq k)=\mathbb{P}(X\gt k)= 1 - \mathbb{P}(X \leq k)$

Additionally, for any $2$ numbers $j$ and $k$ (with $j\lt k$ ), the following properties hold:

$\mathbb{P}(j\lt X\lt k)=\mathbb{P}(j\leq X \lt k)=\mathbb{P}(j \lt X\leq k)=\mathbb{P}(j\leq X\leq k)$
$\mathbb{P}(k\gt X \gt j)=\mathbb{P}(k\geq X \gt j)=\mathbb{P}(k\gt X\geq j)=\mathbb{P}(k \geq X\geq j)$
$\mathbb{P}(j \leq X \leq k) = \mathbb{P}(X \leq k) - \mathbb{P}(X \leq j)$