Statistics Examples

\begin{matrix} x & P (x) 1 & 0.2 3 & 0.2 5 & 0.3 8 & 0.1 10 & 0.2 \end{matrix}

$\begin{array}{c} x & P (x) \\ 1 & 0.2 \\ 3 & 0.2 \\ 5 & 0.3 \\ 8 & 0.1 \\ 10 & 0.2 \end{array}$

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

$x$ takes a set of separate values (such as

0

$0$ ,

1

$1$ ,

2

$2$ ...). Its probability distribution assigns a probability

P (x)

$P (x)$ to each possible value

x

$x$ . For each

x

$x$ , the probability

P (x)

$P (x)$ falls between

0

$0$ and

1

$1$ inclusive and the sum of the probabilities for all the possible

x

$x$ values equals to

1

$1$ .

1. For each

x

$x$ ,

0 \leq P (x) \leq 1

$0 \leq P (x) \leq 1$ .

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

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

Step 1.2

0.2

$0.2$ is between

0

$0$ and

1

$1$ inclusive, which meets the first property of the probability distribution.

0.2

$0.2$ is between

0

$0$ and

1

$1$ inclusive

Step 1.3

0.3

$0.3$ is between

0

$0$ and

1

$1$ inclusive, which meets the first property of the probability distribution.

0.3

$0.3$ is between

0

$0$ and

1

$1$ inclusive

Step 1.4

0.1

$0.1$ is between

0

$0$ and

1

$1$ inclusive, which meets the first property of the probability distribution.

0.1

$0.1$ is between

0

$0$ and

1

$1$ inclusive

Step 1.5

0.2

$0.2$ is between

0

$0$ and

1

$1$ inclusive, which meets the first property of the probability distribution.

0.2

$0.2$ is between

0

$0$ and

1

$1$ inclusive

Step 1.6

For each

x

$x$ , the probability

P (x)

$P (x)$ falls between

0

$0$ and

1

$1$ inclusive, which meets the first property of the probability distribution.

0 \leq P (x) \leq 1

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

Step 1.7

Find the sum of the probabilities for all the possible

x

$x$ values.

0.2 + 0.2 + 0.3 + 0.1 + 0.2

$0.2 + 0.2 + 0.3 + 0.1 + 0.2$

Step 1.8

The sum of the probabilities for all the possible

x

$x$ values is

0.2 + 0.2 + 0.3 + 0.1 + 0.2 = 1

$0.2 + 0.2 + 0.3 + 0.1 + 0.2 = 1$ .

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Step 1.8.1

Add

0.2

$0.2$ and

0.2

$0.2$ .

0.4 + 0.3 + 0.1 + 0.2

$0.4 + 0.3 + 0.1 + 0.2$

Step 1.8.2

Add

0.4

$0.4$ and

0.3

$0.3$ .

0.7 + 0.1 + 0.2

$0.7 + 0.1 + 0.2$

Step 1.8.3

Add

0.7

$0.7$ and

0.1

$0.1$ .

0.8 + 0.2

$0.8 + 0.2$

Step 1.8.4

Add

0.8

$0.8$ and

0.2

$0.2$ .

1

$1$

1

$1$

Step 1.9

For each

x

$x$ , the probability of

P (x)

$P (x)$ falls between

0

$0$ and

1

$1$ inclusive. In addition, the sum of the probabilities for all the possible

x

$x$ equals

1

$1$ , 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

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

x

$x$ values

Property 2:

0.2 + 0.2 + 0.3 + 0.1 + 0.2 = 1

$0.2 + 0.2 + 0.3 + 0.1 + 0.2 = 1$

The table satisfies the two properties of a probability distribution:

Property 1:

0 \leq P (x) \leq 1

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

x

$x$ values

Property 2:

0.2 + 0.2 + 0.3 + 0.1 + 0.2 = 1

$0.2 + 0.2 + 0.3 + 0.1 + 0.2 = 1$

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.

1 \cdot 0.2 + 3 \cdot 0.2 + 5 \cdot 0.3 + 8 \cdot 0.1 + 10 \cdot 0.2

$1 \cdot 0.2 + 3 \cdot 0.2 + 5 \cdot 0.3 + 8 \cdot 0.1 + 10 \cdot 0.2$

Step 3

Simplify each term.

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Step 3.1

Multiply

0.2

$0.2$ by

1

$1$ .

0.2 + 3 \cdot 0.2 + 5 \cdot 0.3 + 8 \cdot 0.1 + 10 \cdot 0.2

$0.2 + 3 \cdot 0.2 + 5 \cdot 0.3 + 8 \cdot 0.1 + 10 \cdot 0.2$

Step 3.2

Multiply

3

$3$ by

0.2

$0.2$ .

0.2 + 0.6 + 5 \cdot 0.3 + 8 \cdot 0.1 + 10 \cdot 0.2

$0.2 + 0.6 + 5 \cdot 0.3 + 8 \cdot 0.1 + 10 \cdot 0.2$

Step 3.3

Multiply

5

$5$ by

0.3

$0.3$ .

0.2 + 0.6 + 1.5 + 8 \cdot 0.1 + 10 \cdot 0.2

$0.2 + 0.6 + 1.5 + 8 \cdot 0.1 + 10 \cdot 0.2$

Step 3.4

Multiply

8

$8$ by

0.1

$0.1$ .

0.2 + 0.6 + 1.5 + 0.8 + 10 \cdot 0.2

$0.2 + 0.6 + 1.5 + 0.8 + 10 \cdot 0.2$

Step 3.5

Multiply

10

$10$ by

0.2

$0.2$ .

0.2 + 0.6 + 1.5 + 0.8 + 2

$0.2 + 0.6 + 1.5 + 0.8 + 2$

0.2 + 0.6 + 1.5 + 0.8 + 2

$0.2 + 0.6 + 1.5 + 0.8 + 2$

Step 4

Simplify by adding numbers.

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Step 4.1

Add

0.2

$0.2$ and

0.6

$0.6$ .

0.8 + 1.5 + 0.8 + 2

$0.8 + 1.5 + 0.8 + 2$

Step 4.2

Add

0.8

$0.8$ and

1.5

$1.5$ .

2.3 + 0.8 + 2

$2.3 + 0.8 + 2$

Step 4.3

Add

2.3

$2.3$ and

0.8

$0.8$ .

3.1 + 2

$3.1 + 2$

Step 4.4

Add

3.1

$3.1$ and

2

$2$ .

5.1

$5.1$

5.1

$5.1$

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))}

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

Step 6

Fill in the known values.

\sqrt{{(1 - (5.1))}^{2} \cdot 0.2 + {(3 - (5.1))}^{2} \cdot 0.2 + {(5 - (5.1))}^{2} \cdot 0.3 + {(8 - (5.1))}^{2} \cdot 0.1 + {(10 - (5.1))}^{2} \cdot 0.2}

$\sqrt{{(1 - (5.1))}^{2} \cdot 0.2 + {(3 - (5.1))}^{2} \cdot 0.2 + {(5 - (5.1))}^{2} \cdot 0.3 + {(8 - (5.1))}^{2} \cdot 0.1 + {(10 - (5.1))}^{2} \cdot 0.2}$

Step 7

Simplify the expression.

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Step 7.1

Multiply

- 1

$- 1$ by

5.1

$5.1$ .

\sqrt{{(1 - 5.1)}^{2} \cdot 0.2 + {(3 - (5.1))}^{2} \cdot 0.2 + {(5 - (5.1))}^{2} \cdot 0.3 + {(8 - (5.1))}^{2} \cdot 0.1 + {(10 - (5.1))}^{2} \cdot 0.2}

$\sqrt{{(1 - 5.1)}^{2} \cdot 0.2 + {(3 - (5.1))}^{2} \cdot 0.2 + {(5 - (5.1))}^{2} \cdot 0.3 + {(8 - (5.1))}^{2} \cdot 0.1 + {(10 - (5.1))}^{2} \cdot 0.2}$

Step 7.2

Subtract

5.1

$5.1$ from

1

$1$ .

\sqrt{{(- 4.1)}^{2} \cdot 0.2 + {(3 - (5.1))}^{2} \cdot 0.2 + {(5 - (5.1))}^{2} \cdot 0.3 + {(8 - (5.1))}^{2} \cdot 0.1 + {(10 - (5.1))}^{2} \cdot 0.2}

$\sqrt{{(- 4.1)}^{2} \cdot 0.2 + {(3 - (5.1))}^{2} \cdot 0.2 + {(5 - (5.1))}^{2} \cdot 0.3 + {(8 - (5.1))}^{2} \cdot 0.1 + {(10 - (5.1))}^{2} \cdot 0.2}$

Step 7.3

Raise

- 4.1

$- 4.1$ to the power of

2

$2$ .

\sqrt{16.81 \cdot 0.2 + {(3 - (5.1))}^{2} \cdot 0.2 + {(5 - (5.1))}^{2} \cdot 0.3 + {(8 - (5.1))}^{2} \cdot 0.1 + {(10 - (5.1))}^{2} \cdot 0.2}

$\sqrt{16.81 \cdot 0.2 + {(3 - (5.1))}^{2} \cdot 0.2 + {(5 - (5.1))}^{2} \cdot 0.3 + {(8 - (5.1))}^{2} \cdot 0.1 + {(10 - (5.1))}^{2} \cdot 0.2}$

Step 7.4

Multiply

$16.81$ by

$0.2$ .

$\sqrt{3.362 + {(3 - (5.1))}^{2} \cdot 0.2 + {(5 - (5.1))}^{2} \cdot 0.3 + {(8 - (5.1))}^{2} \cdot 0.1 + {(10 - (5.1))}^{2} \cdot 0.2}$

Step 7.5

Multiply

$- 1$ by

$5.1$ .

$\sqrt{3.362 + {(3 - 5.1)}^{2} \cdot 0.2 + {(5 - (5.1))}^{2} \cdot 0.3 + {(8 - (5.1))}^{2} \cdot 0.1 + {(10 - (5.1))}^{2} \cdot 0.2}$

Step 7.6

Subtract

$5.1$ from

$3$ .

$\sqrt{3.362 + {(- 2.1)}^{2} \cdot 0.2 + {(5 - (5.1))}^{2} \cdot 0.3 + {(8 - (5.1))}^{2} \cdot 0.1 + {(10 - (5.1))}^{2} \cdot 0.2}$

Step 7.7

Raise

$- 2.1$ to the power of

$2$ .

$\sqrt{3.362 + 4.41 \cdot 0.2 + {(5 - (5.1))}^{2} \cdot 0.3 + {(8 - (5.1))}^{2} \cdot 0.1 + {(10 - (5.1))}^{2} \cdot 0.2}$

Step 7.8

Multiply

$4.41$ by

$0.2$ .

$\sqrt{3.362 + 0.882 + {(5 - (5.1))}^{2} \cdot 0.3 + {(8 - (5.1))}^{2} \cdot 0.1 + {(10 - (5.1))}^{2} \cdot 0.2}$

Step 7.9

Multiply

$- 1$ by

$5.1$ .

$\sqrt{3.362 + 0.882 + {(5 - 5.1)}^{2} \cdot 0.3 + {(8 - (5.1))}^{2} \cdot 0.1 + {(10 - (5.1))}^{2} \cdot 0.2}$

Step 7.10

Subtract

$5.1$ from

$5$ .

$\sqrt{3.362 + 0.882 + {(- 0.1)}^{2} \cdot 0.3 + {(8 - (5.1))}^{2} \cdot 0.1 + {(10 - (5.1))}^{2} \cdot 0.2}$

Step 7.11

Raise

$- 0.1$ to the power of

$2$ .

$\sqrt{3.362 + 0.882 + 0.01 \cdot 0.3 + {(8 - (5.1))}^{2} \cdot 0.1 + {(10 - (5.1))}^{2} \cdot 0.2}$

Step 7.12

Multiply

$0.01$ by

$0.3$ .

$\sqrt{3.362 + 0.882 + 0.003 + {(8 - (5.1))}^{2} \cdot 0.1 + {(10 - (5.1))}^{2} \cdot 0.2}$

Step 7.13

Multiply

$- 1$ by

$5.1$ .

$\sqrt{3.362 + 0.882 + 0.003 + {(8 - 5.1)}^{2} \cdot 0.1 + {(10 - (5.1))}^{2} \cdot 0.2}$

Step 7.14

Subtract

$5.1$ from

$8$ .

$\sqrt{3.362 + 0.882 + 0.003 + {2.9}^{2} \cdot 0.1 + {(10 - (5.1))}^{2} \cdot 0.2}$

Step 7.15

Raise

$2.9$ to the power of

$2$ .

$\sqrt{3.362 + 0.882 + 0.003 + 8.41 \cdot 0.1 + {(10 - (5.1))}^{2} \cdot 0.2}$

Step 7.16

Multiply

$8.41$ by

$0.1$ .

$\sqrt{3.362 + 0.882 + 0.003 + 0.841 + {(10 - (5.1))}^{2} \cdot 0.2}$

Step 7.17

Multiply

$- 1$ by

$5.1$ .

$\sqrt{3.362 + 0.882 + 0.003 + 0.841 + {(10 - 5.1)}^{2} \cdot 0.2}$

Step 7.18

Subtract

$5.1$ from

$10$ .

$\sqrt{3.362 + 0.882 + 0.003 + 0.841 + {4.9}^{2} \cdot 0.2}$

Step 7.19

Raise

$4.9$ to the power of

$2$ .

$\sqrt{3.362 + 0.882 + 0.003 + 0.841 + 24.01 \cdot 0.2}$

Step 7.20

Multiply

$24.01$ by

$0.2$ .

$\sqrt{3.362 + 0.882 + 0.003 + 0.841 + 4.802}$

Step 7.21

Add

$3.362$ and

$0.882$ .

$\sqrt{4.244 + 0.003 + 0.841 + 4.802}$

Step 7.22

Add

$4.244$ and

$0.003$ .

$\sqrt{4.247 + 0.841 + 4.802}$

Step 7.23

Add

$4.247$ and

$0.841$ .

$\sqrt{5.088 + 4.802}$

Step 7.24

Add

$5.088$ and

$4.802$ .

$\sqrt{9.89}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\sqrt{9.89}$

Decimal Form:

$3.14483703 \dots$