Statistics Examples

\begin{array}{c} x & P (x) \\ 3 & 0.1 \\ 6 & 0.2 \\ 8 & 0.3 \\ 11 & 0.2 \\ 15 & 0.2 \end{array}

$\begin{array}{c} x & P (x) \\ 3 & 0.1 \\ 6 & 0.2 \\ 8 & 0.3 \\ 11 & 0.2 \\ 15 & 0.2 \end{array}$

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

0.1

$0.1$ 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 1.3

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 1.4

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 1.5

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.

0.1 + 0.2 + 0.3 + 0.2 + 0.2

$0.1 + 0.2 + 0.3 + 0.2 + 0.2$

Step 1.8

The sum of the probabilities for all the possible

x

$x$ values is

0.1 + 0.2 + 0.3 + 0.2 + 0.2 = 1

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

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Step 1.8.1

Add

0.1

$0.1$ and

0.2

$0.2$ .

0.3 + 0.3 + 0.2 + 0.2

$0.3 + 0.3 + 0.2 + 0.2$

Step 1.8.2

Add

0.3

$0.3$ and

0.3

$0.3$ .

0.6 + 0.2 + 0.2

$0.6 + 0.2 + 0.2$

Step 1.8.3

Add

0.6

$0.6$ and

0.2

$0.2$ .

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:

0.1 + 0.2 + 0.3 + 0.2 + 0.2 = 1

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

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:

0.1 + 0.2 + 0.3 + 0.2 + 0.2 = 1

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

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.

u = 3 \cdot 0.1 + 6 \cdot 0.2 + 8 \cdot 0.3 + 11 \cdot 0.2 + 15 \cdot 0.2

$u = 3 \cdot 0.1 + 6 \cdot 0.2 + 8 \cdot 0.3 + 11 \cdot 0.2 + 15 \cdot 0.2$

Step 3

Simplify each term.

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Step 3.1

Multiply

3

$3$ by

0.1

$0.1$ .

u = 0.3 + 6 \cdot 0.2 + 8 \cdot 0.3 + 11 \cdot 0.2 + 15 \cdot 0.2

$u = 0.3 + 6 \cdot 0.2 + 8 \cdot 0.3 + 11 \cdot 0.2 + 15 \cdot 0.2$

Step 3.2

Multiply

6

$6$ by

0.2

$0.2$ .

u = 0.3 + 1.2 + 8 \cdot 0.3 + 11 \cdot 0.2 + 15 \cdot 0.2

$u = 0.3 + 1.2 + 8 \cdot 0.3 + 11 \cdot 0.2 + 15 \cdot 0.2$

Step 3.3

Multiply

8

$8$ by

0.3

$0.3$ .

u = 0.3 + 1.2 + 2.4 + 11 \cdot 0.2 + 15 \cdot 0.2

$u = 0.3 + 1.2 + 2.4 + 11 \cdot 0.2 + 15 \cdot 0.2$

Step 3.4

Multiply

11

$11$ by

0.2

$0.2$ .

u = 0.3 + 1.2 + 2.4 + 2.2 + 15 \cdot 0.2

$u = 0.3 + 1.2 + 2.4 + 2.2 + 15 \cdot 0.2$

Step 3.5

Multiply

15

$15$ by

0.2

$0.2$ .

u = 0.3 + 1.2 + 2.4 + 2.2 + 3

$u = 0.3 + 1.2 + 2.4 + 2.2 + 3$

u = 0.3 + 1.2 + 2.4 + 2.2 + 3

$u = 0.3 + 1.2 + 2.4 + 2.2 + 3$

Step 4

Simplify by adding numbers.

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Step 4.1

Add

0.3

$0.3$ and

1.2

$1.2$ .

u = 1.5 + 2.4 + 2.2 + 3

$u = 1.5 + 2.4 + 2.2 + 3$

Step 4.2

Add

1.5

$1.5$ and

2.4

$2.4$ .

u = 3.9 + 2.2 + 3

$u = 3.9 + 2.2 + 3$

Step 4.3

Add

3.9

$3.9$ and

2.2

$2.2$ .

u = 6.1 + 3

$u = 6.1 + 3$

Step 4.4

Add

6.1

$6.1$ and

3

$3$ .

u = 9.1

$u = 9.1$

u = 9.1

$u = 9.1$

Step 5

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

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

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

Step 6

Fill in the known values.

{(3 - (9.1))}^{2} \cdot 0.1 + {(6 - (9.1))}^{2} \cdot 0.2 + {(8 - (9.1))}^{2} \cdot 0.3 + {(11 - (9.1))}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2

${(3 - (9.1))}^{2} \cdot 0.1 + {(6 - (9.1))}^{2} \cdot 0.2 + {(8 - (9.1))}^{2} \cdot 0.3 + {(11 - (9.1))}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2$

Step 7

Simplify the expression.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply

- 1

$- 1$ by

9.1

$9.1$ .

{(3 - 9.1)}^{2} \cdot 0.1 + {(6 - (9.1))}^{2} \cdot 0.2 + {(8 - (9.1))}^{2} \cdot 0.3 + {(11 - (9.1))}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2

${(3 - 9.1)}^{2} \cdot 0.1 + {(6 - (9.1))}^{2} \cdot 0.2 + {(8 - (9.1))}^{2} \cdot 0.3 + {(11 - (9.1))}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2$

Step 7.1.2

Subtract

9.1

$9.1$ from

3

$3$ .

{(- 6.1)}^{2} \cdot 0.1 + {(6 - (9.1))}^{2} \cdot 0.2 + {(8 - (9.1))}^{2} \cdot 0.3 + {(11 - (9.1))}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2

${(- 6.1)}^{2} \cdot 0.1 + {(6 - (9.1))}^{2} \cdot 0.2 + {(8 - (9.1))}^{2} \cdot 0.3 + {(11 - (9.1))}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2$

Step 7.1.3

Raise

- 6.1

$- 6.1$ to the power of

2

$2$ .

37.21 \cdot 0.1 + {(6 - (9.1))}^{2} \cdot 0.2 + {(8 - (9.1))}^{2} \cdot 0.3 + {(11 - (9.1))}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2

$37.21 \cdot 0.1 + {(6 - (9.1))}^{2} \cdot 0.2 + {(8 - (9.1))}^{2} \cdot 0.3 + {(11 - (9.1))}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2$

Step 7.1.4

Multiply

37.21

$37.21$ by

0.1

$0.1$ .

3.721 + {(6 - (9.1))}^{2} \cdot 0.2 + {(8 - (9.1))}^{2} \cdot 0.3 + {(11 - (9.1))}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2

$3.721 + {(6 - (9.1))}^{2} \cdot 0.2 + {(8 - (9.1))}^{2} \cdot 0.3 + {(11 - (9.1))}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2$

Step 7.1.5

Multiply

- 1

$- 1$ by

9.1

$9.1$ .

3.721 + {(6 - 9.1)}^{2} \cdot 0.2 + {(8 - (9.1))}^{2} \cdot 0.3 + {(11 - (9.1))}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2

$3.721 + {(6 - 9.1)}^{2} \cdot 0.2 + {(8 - (9.1))}^{2} \cdot 0.3 + {(11 - (9.1))}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2$

Step 7.1.6

Subtract

9.1

$9.1$ from

6

$6$ .

3.721 + {(- 3.1)}^{2} \cdot 0.2 + {(8 - (9.1))}^{2} \cdot 0.3 + {(11 - (9.1))}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2

$3.721 + {(- 3.1)}^{2} \cdot 0.2 + {(8 - (9.1))}^{2} \cdot 0.3 + {(11 - (9.1))}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2$

Step 7.1.7

Raise

- 3.1

$- 3.1$ to the power of

2

$2$ .

3.721 + 9.61 \cdot 0.2 + {(8 - (9.1))}^{2} \cdot 0.3 + {(11 - (9.1))}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2

$3.721 + 9.61 \cdot 0.2 + {(8 - (9.1))}^{2} \cdot 0.3 + {(11 - (9.1))}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2$

Step 7.1.8

Multiply

9.61

$9.61$ by

0.2

$0.2$ .

3.721 + 1.922 + {(8 - (9.1))}^{2} \cdot 0.3 + {(11 - (9.1))}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2

$3.721 + 1.922 + {(8 - (9.1))}^{2} \cdot 0.3 + {(11 - (9.1))}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2$

Step 7.1.9

Multiply

- 1

$- 1$ by

9.1

$9.1$ .

3.721 + 1.922 + {(8 - 9.1)}^{2} \cdot 0.3 + {(11 - (9.1))}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2

$3.721 + 1.922 + {(8 - 9.1)}^{2} \cdot 0.3 + {(11 - (9.1))}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2$

Step 7.1.10

Subtract

9.1

$9.1$ from

8

$8$ .

3.721 + 1.922 + {(- 1.1)}^{2} \cdot 0.3 + {(11 - (9.1))}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2

$3.721 + 1.922 + {(- 1.1)}^{2} \cdot 0.3 + {(11 - (9.1))}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2$

Step 7.1.11

Raise

- 1.1

$- 1.1$ to the power of

2

$2$ .

3.721 + 1.922 + 1.21 \cdot 0.3 + {(11 - (9.1))}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2

$3.721 + 1.922 + 1.21 \cdot 0.3 + {(11 - (9.1))}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2$

Step 7.1.12

Multiply

1.21

$1.21$ by

0.3

$0.3$ .

3.721 + 1.922 + 0.363 + {(11 - (9.1))}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2

$3.721 + 1.922 + 0.363 + {(11 - (9.1))}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2$

Step 7.1.13

Multiply

- 1

$- 1$ by

9.1

$9.1$ .

3.721 + 1.922 + 0.363 + {(11 - 9.1)}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2

$3.721 + 1.922 + 0.363 + {(11 - 9.1)}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2$

Step 7.1.14

Subtract

9.1

$9.1$ from

11

$11$ .

3.721 + 1.922 + 0.363 + {1.9}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2

$3.721 + 1.922 + 0.363 + {1.9}^{2} \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2$

Step 7.1.15

Raise

1.9

$1.9$ to the power of

2

$2$ .

3.721 + 1.922 + 0.363 + 3.61 \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2

$3.721 + 1.922 + 0.363 + 3.61 \cdot 0.2 + {(15 - (9.1))}^{2} \cdot 0.2$

Step 7.1.16

Multiply

3.61

$3.61$ by

0.2

$0.2$ .

3.721 + 1.922 + 0.363 + 0.722 + {(15 - (9.1))}^{2} \cdot 0.2

$3.721 + 1.922 + 0.363 + 0.722 + {(15 - (9.1))}^{2} \cdot 0.2$

Step 7.1.17

Multiply

- 1

$- 1$ by

9.1

$9.1$ .

3.721 + 1.922 + 0.363 + 0.722 + {(15 - 9.1)}^{2} \cdot 0.2

$3.721 + 1.922 + 0.363 + 0.722 + {(15 - 9.1)}^{2} \cdot 0.2$

Step 7.1.18

Subtract

9.1

$9.1$ from

15

$15$ .

3.721 + 1.922 + 0.363 + 0.722 + {5.9}^{2} \cdot 0.2

$3.721 + 1.922 + 0.363 + 0.722 + {5.9}^{2} \cdot 0.2$

Step 7.1.19

Raise

5.9

$5.9$ to the power of

2

$2$ .

3.721 + 1.922 + 0.363 + 0.722 + 34.81 \cdot 0.2

$3.721 + 1.922 + 0.363 + 0.722 + 34.81 \cdot 0.2$

Step 7.1.20

Multiply

34.81

$34.81$ by

0.2

$0.2$ .

3.721 + 1.922 + 0.363 + 0.722 + 6.962

$3.721 + 1.922 + 0.363 + 0.722 + 6.962$

3.721 + 1.922 + 0.363 + 0.722 + 6.962

$3.721 + 1.922 + 0.363 + 0.722 + 6.962$

Step 7.2

Simplify by adding numbers.

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Step 7.2.1

Add

3.721

$3.721$ and

1.922

$1.922$ .

5.643 + 0.363 + 0.722 + 6.962

$5.643 + 0.363 + 0.722 + 6.962$

Step 7.2.2

Add

5.643

$5.643$ and

0.363

$0.363$ .

6.006 + 0.722 + 6.962

$6.006 + 0.722 + 6.962$

Step 7.2.3

Add

6.006

$6.006$ and

0.722

$0.722$ .

6.728 + 6.962

$6.728 + 6.962$

Step 7.2.4

Add

6.728

$6.728$ and

6.962

$6.962$ .

13.69

$13.69$

13.69

$13.69$

13.69

$13.69$

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