Statistics 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 ${\displaystyle 1}$.

${\displaystyle 0.4}$ 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.

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

${\displaystyle 0.1}$ 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 ${\displaystyle 0.4+0.3+0.2+0.1=1}$.

For each ${\displaystyle x}$, the probability of ${\displaystyle P\left(x\right)}$ falls between ${\displaystyle 0}$ and ${\displaystyle 1}$ inclusive. In addition, the sum of the probabilities for all the possible ${\displaystyle x}$ equals ${\displaystyle 1}$, which means that the table satisfies the two properties of a probability distribution.

Multiply ${\displaystyle 3}$ by ${\displaystyle 0.4}$.

Multiply ${\displaystyle 7}$ by ${\displaystyle 0.3}$.

Multiply ${\displaystyle 9}$ by ${\displaystyle 0.2}$.

Multiply ${\displaystyle 10}$ by ${\displaystyle 0.1}$.

Add ${\displaystyle 1.2}$ and ${\displaystyle 2.1}$.

Add ${\displaystyle 3.3}$ and ${\displaystyle 1.8}$.

Add ${\displaystyle 5.1}$ and ${\displaystyle 1}$.

Multiply ${\displaystyle -1}$ by ${\displaystyle 6.1}$.

Subtract ${\displaystyle 6.1}$ from ${\displaystyle 3}$.

Raise ${\displaystyle -3.1}$ to the power of ${\displaystyle 2}$.

Multiply ${\displaystyle 9.61}$ by ${\displaystyle 0.4}$.

Multiply ${\displaystyle -1}$ by ${\displaystyle 6.1}$.

Subtract ${\displaystyle 6.1}$ from ${\displaystyle 7}$.

Raise ${\displaystyle 0.9}$ to the power of ${\displaystyle 2}$.

Multiply ${\displaystyle 0.81}$ by ${\displaystyle 0.3}$.

Multiply ${\displaystyle -1}$ by ${\displaystyle 6.1}$.

Subtract ${\displaystyle 6.1}$ from ${\displaystyle 9}$.

Raise ${\displaystyle 2.9}$ to the power of ${\displaystyle 2}$.

Multiply ${\displaystyle 8.41}$ by ${\displaystyle 0.2}$.

Multiply ${\displaystyle -1}$ by ${\displaystyle 6.1}$.

Subtract ${\displaystyle 6.1}$ from ${\displaystyle 10}$.

Raise ${\displaystyle 3.9}$ to the power of ${\displaystyle 2}$.

Multiply ${\displaystyle 15.21}$ by ${\displaystyle 0.1}$.

Add ${\displaystyle 3.844}$ and ${\displaystyle 0.243}$.

Add ${\displaystyle 4.087}$ and ${\displaystyle 1.682}$.

Add ${\displaystyle 5.769}$ and ${\displaystyle 1.521}$.

Rewrite ${\displaystyle 7.29}$ as ${\displaystyle {2.7}^2}$.

Pull terms out from under the radical, assuming positive real numbers.