Finite Math Examples

$\begin{array}{c} x & P (x) \\ 0 & 0.353 \\ 1 & 0.237 \\ 2 & 0.203 \\ 3 & 0.113 \\ 4 & 0.094 \\ 5 & 0 \\ 6 & 0 \end{array}$

Step 1

Prove that the given table satisfies the two properties needed for a probability distribution.

Tap for more steps...

Step 1.1

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

1. For each $x$ , $0 \leq P (x) \leq 1$ .

2. $P (x_{0}) + P (x_{1}) + P (x_{2}) + \dots + P (x_{n}) = 1$ .

Step 1.2

$0.353$ is between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

$0.353$ is between $0$ and $1$ inclusive

Step 1.3

$0.237$ is between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

$0.237$ is between $0$ and $1$ inclusive

Step 1.4

$0.203$ is between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

$0.203$ is between $0$ and $1$ inclusive

Step 1.5

$0.113$ is between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

$0.113$ is between $0$ and $1$ inclusive

Step 1.6

$0.094$ is between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

$0.094$ is between $0$ and $1$ inclusive

Step 1.7

$0$ is between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

$0$ is between $0$ and $1$ inclusive

Step 1.8

For each $x$ , the probability $P (x)$ falls between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

$0 \leq P (x) \leq 1$ for all x values

Step 1.9

Find the sum of the probabilities for all the possible $x$ values.

$0.353 + 0.237 + 0.203 + 0.113 + 0.094 + 0 + 0$

Step 1.10

The sum of the probabilities for all the possible $x$ values is $0.353 + 0.237 + 0.203 + 0.113 + 0.094 + 0 + 0 = 1$ .

Tap for more steps...

Step 1.10.1

Add $0.353$ and $0.237$ .

$0.59 + 0.203 + 0.113 + 0.094 + 0 + 0$

Step 1.10.2

Add $0.59$ and $0.203$ .

$0.793 + 0.113 + 0.094 + 0 + 0$

Step 1.10.3

Add $0.793$ and $0.113$ .

$0.906 + 0.094 + 0 + 0$

Step 1.10.4

Add $0.906$ and $0.094$ .

$1 + 0 + 0$

Step 1.10.5

Add $1$ and $0$ .

$1 + 0$

Step 1.10.6

Add $1$ and $0$ .

$1$

Step 1.11

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

The table satisfies the two properties of a probability distribution:

Property 1: $0 \leq P (x) \leq 1$ for all $x$ values

Property 2: $0.353 + 0.237 + 0.203 + 0.113 + 0.094 + 0 + 0 = 1$

The table satisfies the two properties of a probability distribution:

Property 1: $0 \leq P (x) \leq 1$ for all $x$ values

Property 2: $0.353 + 0.237 + 0.203 + 0.113 + 0.094 + 0 + 0 = 1$

Step 2

The expectation mean of a distribution is the value expected if trials of the distribution could continue indefinitely. This is equal to each value multiplied by its discrete probability.

$0 \cdot 0.353 + 1 \cdot 0.237 + 2 \cdot 0.203 + 3 \cdot 0.113 + 4 \cdot 0.094 + 5 \cdot 0 + 6 \cdot 0$

Step 3

Simplify each term.

Tap for more steps...

Step 3.1

Multiply $0$ by $0.353$ .

$0 + 1 \cdot 0.237 + 2 \cdot 0.203 + 3 \cdot 0.113 + 4 \cdot 0.094 + 5 \cdot 0 + 6 \cdot 0$

Step 3.2

Multiply $0.237$ by $1$ .

$0 + 0.237 + 2 \cdot 0.203 + 3 \cdot 0.113 + 4 \cdot 0.094 + 5 \cdot 0 + 6 \cdot 0$

Step 3.3

Multiply $2$ by $0.203$ .

$0 + 0.237 + 0.406 + 3 \cdot 0.113 + 4 \cdot 0.094 + 5 \cdot 0 + 6 \cdot 0$

Step 3.4

Multiply $3$ by $0.113$ .

$0 + 0.237 + 0.406 + 0.339 + 4 \cdot 0.094 + 5 \cdot 0 + 6 \cdot 0$

Step 3.5

Multiply $4$ by $0.094$ .

$0 + 0.237 + 0.406 + 0.339 + 0.376 + 5 \cdot 0 + 6 \cdot 0$

Step 3.6

Multiply $5$ by $0$ .

$0 + 0.237 + 0.406 + 0.339 + 0.376 + 0 + 6 \cdot 0$

Step 3.7

Multiply $6$ by $0$ .

$0 + 0.237 + 0.406 + 0.339 + 0.376 + 0 + 0$

Step 4

Simplify by adding numbers.

Tap for more steps...

Step 4.1

Add $0$ and $0.237$ .

$0.237 + 0.406 + 0.339 + 0.376 + 0 + 0$

Step 4.2

Add $0.237$ and $0.406$ .

$0.643 + 0.339 + 0.376 + 0 + 0$

Step 4.3

Add $0.643$ and $0.339$ .

$0.982 + 0.376 + 0 + 0$

Step 4.4

Add $0.982$ and $0.376$ .

$1.358 + 0 + 0$

Step 4.5

Add $1.358$ and $0$ .

$1.358 + 0$

Step 4.6

Add $1.358$ and $0$ .

$1.358$

Step 5

The standard deviation of a distribution is a measure of the dispersion and is equal to the square root of the variance.

$s = \sqrt{\sum_{}^{} {(x - u)}^{2} \cdot (P (x))}$

Step 6

Fill in the known values.

$\sqrt{{(0 - (1.358))}^{2} \cdot 0.353 + {(1 - (1.358))}^{2} \cdot 0.237 + {(2 - (1.358))}^{2} \cdot 0.203 + {(3 - (1.358))}^{2} \cdot 0.113 + {(4 - (1.358))}^{2} \cdot 0.094 + {(5 - (1.358))}^{2} \cdot 0 + {(6 - (1.358))}^{2} \cdot 0}$

Step 7

Simplify the expression.

Tap for more steps...

Step 7.1

Multiply $- 1$ by $1.358$ .

$\sqrt{{(0 - 1.358)}^{2} \cdot 0.353 + {(1 - (1.358))}^{2} \cdot 0.237 + {(2 - (1.358))}^{2} \cdot 0.203 + {(3 - (1.358))}^{2} \cdot 0.113 + {(4 - (1.358))}^{2} \cdot 0.094 + {(5 - (1.358))}^{2} \cdot 0 + {(6 - (1.358))}^{2} \cdot 0}$

Step 7.2

Subtract $1.358$ from $0$ .

$\sqrt{{(- 1.358)}^{2} \cdot 0.353 + {(1 - (1.358))}^{2} \cdot 0.237 + {(2 - (1.358))}^{2} \cdot 0.203 + {(3 - (1.358))}^{2} \cdot 0.113 + {(4 - (1.358))}^{2} \cdot 0.094 + {(5 - (1.358))}^{2} \cdot 0 + {(6 - (1.358))}^{2} \cdot 0}$

Step 7.3

Raise $- 1.358$ to the power of $2$ .

$\sqrt{1.844164 \cdot 0.353 + {(1 - (1.358))}^{2} \cdot 0.237 + {(2 - (1.358))}^{2} \cdot 0.203 + {(3 - (1.358))}^{2} \cdot 0.113 + {(4 - (1.358))}^{2} \cdot 0.094 + {(5 - (1.358))}^{2} \cdot 0 + {(6 - (1.358))}^{2} \cdot 0}$

Step 7.4

Multiply $1.844164$ by $0.353$ .

$\sqrt{0.650\overline{98} + {(1 - (1.358))}^{2} \cdot 0.237 + {(2 - (1.358))}^{2} \cdot 0.203 + {(3 - (1.358))}^{2} \cdot 0.113 + {(4 - (1.358))}^{2} \cdot 0.094 + {(5 - (1.358))}^{2} \cdot 0 + {(6 - (1.358))}^{2} \cdot 0}$

Step 7.5

Multiply $- 1$ by $1.358$ .

$\sqrt{0.650\overline{98} + {(1 - 1.358)}^{2} \cdot 0.237 + {(2 - (1.358))}^{2} \cdot 0.203 + {(3 - (1.358))}^{2} \cdot 0.113 + {(4 - (1.358))}^{2} \cdot 0.094 + {(5 - (1.358))}^{2} \cdot 0 + {(6 - (1.358))}^{2} \cdot 0}$

Step 7.6

Subtract $1.358$ from $1$ .

$\sqrt{0.650\overline{98} + {(- 0.358)}^{2} \cdot 0.237 + {(2 - (1.358))}^{2} \cdot 0.203 + {(3 - (1.358))}^{2} \cdot 0.113 + {(4 - (1.358))}^{2} \cdot 0.094 + {(5 - (1.358))}^{2} \cdot 0 + {(6 - (1.358))}^{2} \cdot 0}$

Step 7.7

Raise $- 0.358$ to the power of $2$ .

$\sqrt{0.650\overline{98} + 0.128164 \cdot 0.237 + {(2 - (1.358))}^{2} \cdot 0.203 + {(3 - (1.358))}^{2} \cdot 0.113 + {(4 - (1.358))}^{2} \cdot 0.094 + {(5 - (1.358))}^{2} \cdot 0 + {(6 - (1.358))}^{2} \cdot 0}$

Step 7.8

Multiply $0.128164$ by $0.237$ .

$\sqrt{0.650\overline{98} + 0.03037486 + {(2 - (1.358))}^{2} \cdot 0.203 + {(3 - (1.358))}^{2} \cdot 0.113 + {(4 - (1.358))}^{2} \cdot 0.094 + {(5 - (1.358))}^{2} \cdot 0 + {(6 - (1.358))}^{2} \cdot 0}$

Step 7.9

Multiply $- 1$ by $1.358$ .

$\sqrt{0.650\overline{98} + 0.03037486 + {(2 - 1.358)}^{2} \cdot 0.203 + {(3 - (1.358))}^{2} \cdot 0.113 + {(4 - (1.358))}^{2} \cdot 0.094 + {(5 - (1.358))}^{2} \cdot 0 + {(6 - (1.358))}^{2} \cdot 0}$

Step 7.10

Subtract $1.358$ from $2$ .

$\sqrt{0.650\overline{98} + 0.03037486 + {0.642}^{2} \cdot 0.203 + {(3 - (1.358))}^{2} \cdot 0.113 + {(4 - (1.358))}^{2} \cdot 0.094 + {(5 - (1.358))}^{2} \cdot 0 + {(6 - (1.358))}^{2} \cdot 0}$

Step 7.11

Raise $0.642$ to the power of $2$ .

$\sqrt{0.650\overline{98} + 0.03037486 + 0.412164 \cdot 0.203 + {(3 - (1.358))}^{2} \cdot 0.113 + {(4 - (1.358))}^{2} \cdot 0.094 + {(5 - (1.358))}^{2} \cdot 0 + {(6 - (1.358))}^{2} \cdot 0}$

Step 7.12

Multiply $0.412164$ by $0.203$ .

$\sqrt{0.650\overline{98} + 0.03037486 + 0.08366929 + {(3 - (1.358))}^{2} \cdot 0.113 + {(4 - (1.358))}^{2} \cdot 0.094 + {(5 - (1.358))}^{2} \cdot 0 + {(6 - (1.358))}^{2} \cdot 0}$

Step 7.13

Multiply $- 1$ by $1.358$ .

$\sqrt{0.650\overline{98} + 0.03037486 + 0.08366929 + {(3 - 1.358)}^{2} \cdot 0.113 + {(4 - (1.358))}^{2} \cdot 0.094 + {(5 - (1.358))}^{2} \cdot 0 + {(6 - (1.358))}^{2} \cdot 0}$

Step 7.14

Subtract $1.358$ from $3$ .

$\sqrt{0.650\overline{98} + 0.03037486 + 0.08366929 + {1.642}^{2} \cdot 0.113 + {(4 - (1.358))}^{2} \cdot 0.094 + {(5 - (1.358))}^{2} \cdot 0 + {(6 - (1.358))}^{2} \cdot 0}$

Step 7.15

Raise $1.642$ to the power of $2$ .

$\sqrt{0.650\overline{98} + 0.03037486 + 0.08366929 + 2.696164 \cdot 0.113 + {(4 - (1.358))}^{2} \cdot 0.094 + {(5 - (1.358))}^{2} \cdot 0 + {(6 - (1.358))}^{2} \cdot 0}$

Step 7.16

Multiply $2.696164$ by $0.113$ .

$\sqrt{0.650\overline{98} + 0.03037486 + 0.08366929 + 0.30466653 + {(4 - (1.358))}^{2} \cdot 0.094 + {(5 - (1.358))}^{2} \cdot 0 + {(6 - (1.358))}^{2} \cdot 0}$

Step 7.17

Multiply $- 1$ by $1.358$ .

$\sqrt{0.650\overline{98} + 0.03037486 + 0.08366929 + 0.30466653 + {(4 - 1.358)}^{2} \cdot 0.094 + {(5 - (1.358))}^{2} \cdot 0 + {(6 - (1.358))}^{2} \cdot 0}$

Step 7.18

Subtract $1.358$ from $4$ .

$\sqrt{0.650\overline{98} + 0.03037486 + 0.08366929 + 0.30466653 + {2.642}^{2} \cdot 0.094 + {(5 - (1.358))}^{2} \cdot 0 + {(6 - (1.358))}^{2} \cdot 0}$

Step 7.19

Raise $2.642$ to the power of $2$ .

$\sqrt{0.650\overline{98} + 0.03037486 + 0.08366929 + 0.30466653 + 6.980164 \cdot 0.094 + {(5 - (1.358))}^{2} \cdot 0 + {(6 - (1.358))}^{2} \cdot 0}$

Step 7.20

Multiply $6.980164$ by $0.094$ .

$\sqrt{0.650\overline{98} + 0.03037486 + 0.08366929 + 0.30466653 + 0.65613541 + {(5 - (1.358))}^{2} \cdot 0 + {(6 - (1.358))}^{2} \cdot 0}$

Step 7.21

Multiply $- 1$ by $1.358$ .

$\sqrt{0.650\overline{98} + 0.03037486 + 0.08366929 + 0.30466653 + 0.65613541 + {(5 - 1.358)}^{2} \cdot 0 + {(6 - (1.358))}^{2} \cdot 0}$

Step 7.22

Subtract $1.358$ from $5$ .

$\sqrt{0.650\overline{98} + 0.03037486 + 0.08366929 + 0.30466653 + 0.65613541 + {3.642}^{2} \cdot 0 + {(6 - (1.358))}^{2} \cdot 0}$

Step 7.23

Raise $3.642$ to the power of $2$ .

$\sqrt{0.650\overline{98} + 0.03037486 + 0.08366929 + 0.30466653 + 0.65613541 + 13.264164 \cdot 0 + {(6 - (1.358))}^{2} \cdot 0}$

Step 7.24

Multiply $13.264164$ by $0$ .

$\sqrt{0.650\overline{98} + 0.03037486 + 0.08366929 + 0.30466653 + 0.65613541 + 0 + {(6 - (1.358))}^{2} \cdot 0}$

Step 7.25

Multiply $- 1$ by $1.358$ .

$\sqrt{0.650\overline{98} + 0.03037486 + 0.08366929 + 0.30466653 + 0.65613541 + 0 + {(6 - 1.358)}^{2} \cdot 0}$

Step 7.26

Subtract $1.358$ from $6$ .

$\sqrt{0.650\overline{98} + 0.03037486 + 0.08366929 + 0.30466653 + 0.65613541 + 0 + {4.642}^{2} \cdot 0}$

Step 7.27

Raise $4.642$ to the power of $2$ .

$\sqrt{0.650\overline{98} + 0.03037486 + 0.08366929 + 0.30466653 + 0.65613541 + 0 + 21.548164 \cdot 0}$

Step 7.28

Multiply $21.548164$ by $0$ .

$\sqrt{0.650\overline{98} + 0.03037486 + 0.08366929 + 0.30466653 + 0.65613541 + 0 + 0}$

Step 7.29

Simplify by adding zeros.

Tap for more steps...

Step 7.29.1

Add $0.650\overline{98} + 0.03037486 + 0.08366929 + 0.30466653 + 0.65613541 + 0$ and $0$ .

$\sqrt{0.650\overline{98} + 0.03037486 + 0.08366929 + 0.30466653 + 0.65613541 + 0}$

Step 7.29.2

Add $0.650\overline{98} + 0.03037486 + 0.08366929 + 0.30466653 + 0.65613541$ and $0$ .

$\sqrt{0.650\overline{98} + 0.03037486 + 0.08366929 + 0.30466653 + 0.65613541}$

Step 7.30

Add $0.650\overline{98}$ and $0.03037486$ .

$\sqrt{0.68136476 + 0.08366929 + 0.30466653 + 0.65613541}$

Step 7.31

Add $0.68136476$ and $0.08366929$ .

$\sqrt{0.76503405 + 0.30466653 + 0.65613541}$

Step 7.32

Add $0.76503405$ and $0.30466653$ .

$\sqrt{1.06970058 + 0.65613541}$

Step 7.33

Add $1.06970058$ and $0.65613541$ .

$\sqrt{1.725836}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\sqrt{1.725836}$

Decimal Form:

$1.31371077 \dots$