Finite Math Examples Statistical Distributions

Find the Standard Deviation

$\begin{array}{c} x & P (x) \\ 4 & 0.4 \\ 7 & 0.2 \\ 9 & 0.1 \\ 11 & 0.1 \\ 13 & 0.1 \\ 17 & 0.1 \end{array}$

Step 1

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

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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

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

$0.4$ is between $0$ and $1$ inclusive

Step 1.3

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

$0.2$ is between $0$ and $1$ inclusive

Step 1.4

$0.1$ is between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

$0.1$ is between $0$ and $1$ inclusive

Step 1.5

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 1.6

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

$0.4 + 0.2 + 0.1 + 0.1 + 0.1 + 0.1$

Step 1.7

The sum of the probabilities for all the possible $x$ values is $0.4 + 0.2 + 0.1 + 0.1 + 0.1 + 0.1 = 1$ .

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Step 1.7.1

Add $0.4$ and $0.2$ .

$0.6 + 0.1 + 0.1 + 0.1 + 0.1$

Step 1.7.2

Add $0.6$ and $0.1$ .

$0.7 + 0.1 + 0.1 + 0.1$

Step 1.7.3

Add $0.7$ and $0.1$ .

$0.8 + 0.1 + 0.1$

Step 1.7.4

Add $0.8$ and $0.1$ .

$0.9 + 0.1$

Step 1.7.5

Add $0.9$ and $0.1$ .

$1$

Step 1.8

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: $0.4 + 0.2 + 0.1 + 0.1 + 0.1 + 0.1 = 1$

The table satisfies the two properties of a probability distribution:

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

Property 2: $0.4 + 0.2 + 0.1 + 0.1 + 0.1 + 0.1 = 1$

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.

$4 \cdot 0.4 + 7 \cdot 0.2 + 9 \cdot 0.1 + 11 \cdot 0.1 + 13 \cdot 0.1 + 17 \cdot 0.1$

Step 3

Simplify each term.

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Step 3.1

Multiply $4$ by $0.4$ .

$1.6 + 7 \cdot 0.2 + 9 \cdot 0.1 + 11 \cdot 0.1 + 13 \cdot 0.1 + 17 \cdot 0.1$

Step 3.2

Multiply $7$ by $0.2$ .

$1.6 + 1.4 + 9 \cdot 0.1 + 11 \cdot 0.1 + 13 \cdot 0.1 + 17 \cdot 0.1$

Step 3.3

Multiply $9$ by $0.1$ .

$1.6 + 1.4 + 0.9 + 11 \cdot 0.1 + 13 \cdot 0.1 + 17 \cdot 0.1$

Step 3.4

Multiply $11$ by $0.1$ .

$1.6 + 1.4 + 0.9 + 1.1 + 13 \cdot 0.1 + 17 \cdot 0.1$

Step 3.5

Multiply $13$ by $0.1$ .

$1.6 + 1.4 + 0.9 + 1.1 + 1.3 + 17 \cdot 0.1$

Step 3.6

Multiply $17$ by $0.1$ .

$1.6 + 1.4 + 0.9 + 1.1 + 1.3 + 1.7$

Step 4

Simplify by adding numbers.

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Step 4.1

Add $1.6$ and $1.4$ .

$3 + 0.9 + 1.1 + 1.3 + 1.7$

Step 4.2

Add $3$ and $0.9$ .

$3.9 + 1.1 + 1.3 + 1.7$

Step 4.3

Add $3.9$ and $1.1$ .

$5 + 1.3 + 1.7$

Step 4.4

Add $5$ and $1.3$ .

$6.3 + 1.7$

Step 4.5

Add $6.3$ and $1.7$ .

$8$

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.

$\sqrt{{(4 - (8))}^{2} \cdot 0.4 + {(7 - (8))}^{2} \cdot 0.2 + {(9 - (8))}^{2} \cdot 0.1 + {(11 - (8))}^{2} \cdot 0.1 + {(13 - (8))}^{2} \cdot 0.1 + {(17 - (8))}^{2} \cdot 0.1}$

Step 7

Simplify the expression.

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Step 7.1

Multiply $- 1$ by $8$ .

$\sqrt{{(4 - 8)}^{2} \cdot 0.4 + {(7 - (8))}^{2} \cdot 0.2 + {(9 - (8))}^{2} \cdot 0.1 + {(11 - (8))}^{2} \cdot 0.1 + {(13 - (8))}^{2} \cdot 0.1 + {(17 - (8))}^{2} \cdot 0.1}$

Step 7.2

Subtract $8$ from $4$ .

$\sqrt{{(- 4)}^{2} \cdot 0.4 + {(7 - (8))}^{2} \cdot 0.2 + {(9 - (8))}^{2} \cdot 0.1 + {(11 - (8))}^{2} \cdot 0.1 + {(13 - (8))}^{2} \cdot 0.1 + {(17 - (8))}^{2} \cdot 0.1}$

Step 7.3

Raise $- 4$ to the power of $2$ .

$\sqrt{16 \cdot 0.4 + {(7 - (8))}^{2} \cdot 0.2 + {(9 - (8))}^{2} \cdot 0.1 + {(11 - (8))}^{2} \cdot 0.1 + {(13 - (8))}^{2} \cdot 0.1 + {(17 - (8))}^{2} \cdot 0.1}$

Step 7.4

Multiply $16$ by $0.4$ .

$\sqrt{6.4 + {(7 - (8))}^{2} \cdot 0.2 + {(9 - (8))}^{2} \cdot 0.1 + {(11 - (8))}^{2} \cdot 0.1 + {(13 - (8))}^{2} \cdot 0.1 + {(17 - (8))}^{2} \cdot 0.1}$

Step 7.5

Multiply $- 1$ by $8$ .

$\sqrt{6.4 + {(7 - 8)}^{2} \cdot 0.2 + {(9 - (8))}^{2} \cdot 0.1 + {(11 - (8))}^{2} \cdot 0.1 + {(13 - (8))}^{2} \cdot 0.1 + {(17 - (8))}^{2} \cdot 0.1}$

Step 7.6

Subtract $8$ from $7$ .

$\sqrt{6.4 + {(- 1)}^{2} \cdot 0.2 + {(9 - (8))}^{2} \cdot 0.1 + {(11 - (8))}^{2} \cdot 0.1 + {(13 - (8))}^{2} \cdot 0.1 + {(17 - (8))}^{2} \cdot 0.1}$

Step 7.7

Raise $- 1$ to the power of $2$ .

$\sqrt{6.4 + 1 \cdot 0.2 + {(9 - (8))}^{2} \cdot 0.1 + {(11 - (8))}^{2} \cdot 0.1 + {(13 - (8))}^{2} \cdot 0.1 + {(17 - (8))}^{2} \cdot 0.1}$

Step 7.8

Multiply $0.2$ by $1$ .

$\sqrt{6.4 + 0.2 + {(9 - (8))}^{2} \cdot 0.1 + {(11 - (8))}^{2} \cdot 0.1 + {(13 - (8))}^{2} \cdot 0.1 + {(17 - (8))}^{2} \cdot 0.1}$

Step 7.9

Multiply $- 1$ by $8$ .

$\sqrt{6.4 + 0.2 + {(9 - 8)}^{2} \cdot 0.1 + {(11 - (8))}^{2} \cdot 0.1 + {(13 - (8))}^{2} \cdot 0.1 + {(17 - (8))}^{2} \cdot 0.1}$

Step 7.10

Subtract $8$ from $9$ .

$\sqrt{6.4 + 0.2 + 1^{2} \cdot 0.1 + {(11 - (8))}^{2} \cdot 0.1 + {(13 - (8))}^{2} \cdot 0.1 + {(17 - (8))}^{2} \cdot 0.1}$

Step 7.11

One to any power is one.

$\sqrt{6.4 + 0.2 + 1 \cdot 0.1 + {(11 - (8))}^{2} \cdot 0.1 + {(13 - (8))}^{2} \cdot 0.1 + {(17 - (8))}^{2} \cdot 0.1}$

Step 7.12

Multiply $0.1$ by $1$ .

$\sqrt{6.4 + 0.2 + 0.1 + {(11 - (8))}^{2} \cdot 0.1 + {(13 - (8))}^{2} \cdot 0.1 + {(17 - (8))}^{2} \cdot 0.1}$

Step 7.13

Multiply $- 1$ by $8$ .

$\sqrt{6.4 + 0.2 + 0.1 + {(11 - 8)}^{2} \cdot 0.1 + {(13 - (8))}^{2} \cdot 0.1 + {(17 - (8))}^{2} \cdot 0.1}$

Step 7.14

Subtract $8$ from $11$ .

$\sqrt{6.4 + 0.2 + 0.1 + 3^{2} \cdot 0.1 + {(13 - (8))}^{2} \cdot 0.1 + {(17 - (8))}^{2} \cdot 0.1}$

Step 7.15

Raise $3$ to the power of $2$ .

$\sqrt{6.4 + 0.2 + 0.1 + 9 \cdot 0.1 + {(13 - (8))}^{2} \cdot 0.1 + {(17 - (8))}^{2} \cdot 0.1}$

Step 7.16

Multiply $9$ by $0.1$ .

$\sqrt{6.4 + 0.2 + 0.1 + 0.9 + {(13 - (8))}^{2} \cdot 0.1 + {(17 - (8))}^{2} \cdot 0.1}$

Step 7.17

Multiply $- 1$ by $8$ .

$\sqrt{6.4 + 0.2 + 0.1 + 0.9 + {(13 - 8)}^{2} \cdot 0.1 + {(17 - (8))}^{2} \cdot 0.1}$

Step 7.18

Subtract $8$ from $13$ .

$\sqrt{6.4 + 0.2 + 0.1 + 0.9 + 5^{2} \cdot 0.1 + {(17 - (8))}^{2} \cdot 0.1}$

Step 7.19

Raise $5$ to the power of $2$ .

$\sqrt{6.4 + 0.2 + 0.1 + 0.9 + 25 \cdot 0.1 + {(17 - (8))}^{2} \cdot 0.1}$

Step 7.20

Multiply $25$ by $0.1$ .

$\sqrt{6.4 + 0.2 + 0.1 + 0.9 + 2.5 + {(17 - (8))}^{2} \cdot 0.1}$

Step 7.21

Multiply $- 1$ by $8$ .

$\sqrt{6.4 + 0.2 + 0.1 + 0.9 + 2.5 + {(17 - 8)}^{2} \cdot 0.1}$

Step 7.22

Subtract $8$ from $17$ .

$\sqrt{6.4 + 0.2 + 0.1 + 0.9 + 2.5 + 9^{2} \cdot 0.1}$

Step 7.23

Raise $9$ to the power of $2$ .

$\sqrt{6.4 + 0.2 + 0.1 + 0.9 + 2.5 + 81 \cdot 0.1}$

Step 7.24

Multiply $81$ by $0.1$ .

$\sqrt{6.4 + 0.2 + 0.1 + 0.9 + 2.5 + 8.1}$

Step 7.25

Add $6.4$ and $0.2$ .

$\sqrt{6.6 + 0.1 + 0.9 + 2.5 + 8.1}$

Step 7.26

Add $6.6$ and $0.1$ .

$\sqrt{6.7 + 0.9 + 2.5 + 8.1}$

Step 7.27

Add $6.7$ and $0.9$ .

$\sqrt{7.6 + 2.5 + 8.1}$

Step 7.28

Add $7.6$ and $2.5$ .

$\sqrt{10.1 + 8.1}$

Step 7.29

Add $10.1$ and $8.1$ .

$\sqrt{18.2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\sqrt{18.2}$

Decimal Form:

$4.26614580 \dots$

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