Finite Math Examples

A discrete random variable ${\displaystyle x}$ takes a set of separate values (such as ${\displaystyle 0}$, ${\displaystyle 1}$, ${\displaystyle 2}$...). Its probability distribution assigns a probability ${\displaystyle P\left(x\right)}$ to each possible value ${\displaystyle x}$. For each ${\displaystyle x}$, the probability ${\displaystyle P\left(x\right)}$ falls between ${\displaystyle 0}$ and ${\displaystyle 1}$ inclusive and the sum of the probabilities for all the possible ${\displaystyle x}$ values equals to ${\displaystyle 1}$.

${\displaystyle 0.2}$ is between ${\displaystyle 0}$ and ${\displaystyle 1}$ inclusive, which meets the first property of the probability distribution.

${\displaystyle 0.3}$ is between ${\displaystyle 0}$ and ${\displaystyle 1}$ inclusive, which meets the first property of the probability distribution.

${\displaystyle 0.4}$ is between ${\displaystyle 0}$ and ${\displaystyle 1}$ inclusive, which meets the first property of the probability distribution.

For each ${\displaystyle x}$, the probability ${\displaystyle P\left(x\right)}$ falls between ${\displaystyle 0}$ and ${\displaystyle 1}$ inclusive, which meets the first property of the probability distribution.

Find the sum of the probabilities for all the possible ${\displaystyle x}$ values.

The sum of the probabilities for all the possible ${\displaystyle x}$ values is ${\displaystyle 0.2+0.2+0.3+0.4=1.1}$.

The sum of the probabilities for all the possible ${\displaystyle x}$ values is not equal to ${\displaystyle 1}$, which does not meet the second property of the probability distribution.

For each ${\displaystyle x}$, the probability ${\displaystyle P\left(x\right)}$ falls between ${\displaystyle 0}$ and ${\displaystyle 1}$ inclusive. However, the sum of the probabilities for all the possible ${\displaystyle x}$ values is not equal to ${\displaystyle 1}$, which means that the table does not satisfy the two properties of a probability distribution.