Statistics Examples

\begin{matrix} x & P (x) 7 & 0.2 11 & 0.1 14 & 0.3 17 & 0.2 20 & 0.2 \end{matrix}

$\begin{array}{c} x & P (x) \\ 7 & 0.2 \\ 11 & 0.1 \\ 14 & 0.3 \\ 17 & 0.2 \\ 20 & 0.2 \end{array}$

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

0.1

$0.1$ 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 1.4

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.

0.2 + 0.1 + 0.3 + 0.2 + 0.2

$0.2 + 0.1 + 0.3 + 0.2 + 0.2$

Step 1.8

The sum of the probabilities for all the possible

x

$x$ values is

0.2 + 0.1 + 0.3 + 0.2 + 0.2 = 1

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

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Step 1.8.1

Add

0.2

$0.2$ and

0.1

$0.1$ .

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:

0.2 + 0.1 + 0.3 + 0.2 + 0.2 = 1

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

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:

0.2 + 0.1 + 0.3 + 0.2 + 0.2 = 1

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

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 = 7 \cdot 0.2 + 11 \cdot 0.1 + 14 \cdot 0.3 + 17 \cdot 0.2 + 20 \cdot 0.2

$u = 7 \cdot 0.2 + 11 \cdot 0.1 + 14 \cdot 0.3 + 17 \cdot 0.2 + 20 \cdot 0.2$

Step 3

Simplify each term.

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Step 3.1

Multiply

7

$7$ by

0.2

$0.2$ .

u = 1.4 + 11 \cdot 0.1 + 14 \cdot 0.3 + 17 \cdot 0.2 + 20 \cdot 0.2

$u = 1.4 + 11 \cdot 0.1 + 14 \cdot 0.3 + 17 \cdot 0.2 + 20 \cdot 0.2$

Step 3.2

Multiply

11

$11$ by

0.1

$0.1$ .

u = 1.4 + 1.1 + 14 \cdot 0.3 + 17 \cdot 0.2 + 20 \cdot 0.2

$u = 1.4 + 1.1 + 14 \cdot 0.3 + 17 \cdot 0.2 + 20 \cdot 0.2$

Step 3.3

Multiply

14

$14$ by

0.3

$0.3$ .

u = 1.4 + 1.1 + 4.2 + 17 \cdot 0.2 + 20 \cdot 0.2

$u = 1.4 + 1.1 + 4.2 + 17 \cdot 0.2 + 20 \cdot 0.2$

Step 3.4

Multiply

17

$17$ by

0.2

$0.2$ .

u = 1.4 + 1.1 + 4.2 + 3.4 + 20 \cdot 0.2

$u = 1.4 + 1.1 + 4.2 + 3.4 + 20 \cdot 0.2$

Step 3.5

Multiply

20

$20$ by

0.2

$0.2$ .

u = 1.4 + 1.1 + 4.2 + 3.4 + 4

$u = 1.4 + 1.1 + 4.2 + 3.4 + 4$

u = 1.4 + 1.1 + 4.2 + 3.4 + 4

$u = 1.4 + 1.1 + 4.2 + 3.4 + 4$

Step 4

Simplify by adding numbers.

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Step 4.1

Add

1.4

$1.4$ and

1.1

$1.1$ .

u = 2.5 + 4.2 + 3.4 + 4

$u = 2.5 + 4.2 + 3.4 + 4$

Step 4.2

Add

2.5

$2.5$ and

4.2

$4.2$ .

u = 6.7 + 3.4 + 4

$u = 6.7 + 3.4 + 4$

Step 4.3

Add

6.7

$6.7$ and

3.4

$3.4$ .

u = 10.1 + 4

$u = 10.1 + 4$

Step 4.4

Add

10.1

$10.1$ and

4

$4$ .

u = 14.1

$u = 14.1$

u = 14.1

$u = 14.1$

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.

{(7 - (14.1))}^{2} \cdot 0.2 + {(11 - (14.1))}^{2} \cdot 0.1 + {(14 - (14.1))}^{2} \cdot 0.3 + {(17 - (14.1))}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2

${(7 - (14.1))}^{2} \cdot 0.2 + {(11 - (14.1))}^{2} \cdot 0.1 + {(14 - (14.1))}^{2} \cdot 0.3 + {(17 - (14.1))}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2$

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

14.1

$14.1$ .

{(7 - 14.1)}^{2} \cdot 0.2 + {(11 - (14.1))}^{2} \cdot 0.1 + {(14 - (14.1))}^{2} \cdot 0.3 + {(17 - (14.1))}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2

${(7 - 14.1)}^{2} \cdot 0.2 + {(11 - (14.1))}^{2} \cdot 0.1 + {(14 - (14.1))}^{2} \cdot 0.3 + {(17 - (14.1))}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2$

Step 7.1.2

Subtract

14.1

$14.1$ from

7

$7$ .

{(- 7.1)}^{2} \cdot 0.2 + {(11 - (14.1))}^{2} \cdot 0.1 + {(14 - (14.1))}^{2} \cdot 0.3 + {(17 - (14.1))}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2

${(- 7.1)}^{2} \cdot 0.2 + {(11 - (14.1))}^{2} \cdot 0.1 + {(14 - (14.1))}^{2} \cdot 0.3 + {(17 - (14.1))}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2$

Step 7.1.3

Raise

- 7.1

$- 7.1$ to the power of

2

$2$ .

50.41 \cdot 0.2 + {(11 - (14.1))}^{2} \cdot 0.1 + {(14 - (14.1))}^{2} \cdot 0.3 + {(17 - (14.1))}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2

$50.41 \cdot 0.2 + {(11 - (14.1))}^{2} \cdot 0.1 + {(14 - (14.1))}^{2} \cdot 0.3 + {(17 - (14.1))}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2$

Step 7.1.4

Multiply

50.41

$50.41$ by

0.2

$0.2$ .

10.082 + {(11 - (14.1))}^{2} \cdot 0.1 + {(14 - (14.1))}^{2} \cdot 0.3 + {(17 - (14.1))}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2

$10.082 + {(11 - (14.1))}^{2} \cdot 0.1 + {(14 - (14.1))}^{2} \cdot 0.3 + {(17 - (14.1))}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2$

Step 7.1.5

Multiply

- 1

$- 1$ by

14.1

$14.1$ .

10.082 + {(11 - 14.1)}^{2} \cdot 0.1 + {(14 - (14.1))}^{2} \cdot 0.3 + {(17 - (14.1))}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2

$10.082 + {(11 - 14.1)}^{2} \cdot 0.1 + {(14 - (14.1))}^{2} \cdot 0.3 + {(17 - (14.1))}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2$

Step 7.1.6

Subtract

14.1

$14.1$ from

11

$11$ .

10.082 + {(- 3.1)}^{2} \cdot 0.1 + {(14 - (14.1))}^{2} \cdot 0.3 + {(17 - (14.1))}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2

$10.082 + {(- 3.1)}^{2} \cdot 0.1 + {(14 - (14.1))}^{2} \cdot 0.3 + {(17 - (14.1))}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2$

Step 7.1.7

Raise

- 3.1

$- 3.1$ to the power of

2

$2$ .

10.082 + 9.61 \cdot 0.1 + {(14 - (14.1))}^{2} \cdot 0.3 + {(17 - (14.1))}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2

$10.082 + 9.61 \cdot 0.1 + {(14 - (14.1))}^{2} \cdot 0.3 + {(17 - (14.1))}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2$

Step 7.1.8

Multiply

9.61

$9.61$ by

0.1

$0.1$ .

10.082 + 0.961 + {(14 - (14.1))}^{2} \cdot 0.3 + {(17 - (14.1))}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2

$10.082 + 0.961 + {(14 - (14.1))}^{2} \cdot 0.3 + {(17 - (14.1))}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2$

Step 7.1.9

Multiply

- 1

$- 1$ by

14.1

$14.1$ .

10.082 + 0.961 + {(14 - 14.1)}^{2} \cdot 0.3 + {(17 - (14.1))}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2

$10.082 + 0.961 + {(14 - 14.1)}^{2} \cdot 0.3 + {(17 - (14.1))}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2$

Step 7.1.10

Subtract

14.1

$14.1$ from

14

$14$ .

10.082 + 0.961 + {(- 0.1)}^{2} \cdot 0.3 + {(17 - (14.1))}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2

$10.082 + 0.961 + {(- 0.1)}^{2} \cdot 0.3 + {(17 - (14.1))}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2$

Step 7.1.11

Raise

- 0.1

$- 0.1$ to the power of

2

$2$ .

10.082 + 0.961 + 0.01 \cdot 0.3 + {(17 - (14.1))}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2

$10.082 + 0.961 + 0.01 \cdot 0.3 + {(17 - (14.1))}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2$

Step 7.1.12

Multiply

0.01

$0.01$ by

0.3

$0.3$ .

10.082 + 0.961 + 0.003 + {(17 - (14.1))}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2

$10.082 + 0.961 + 0.003 + {(17 - (14.1))}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2$

Step 7.1.13

Multiply

- 1

$- 1$ by

14.1

$14.1$ .

10.082 + 0.961 + 0.003 + {(17 - 14.1)}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2

$10.082 + 0.961 + 0.003 + {(17 - 14.1)}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2$

Step 7.1.14

Subtract

14.1

$14.1$ from

17

$17$ .

10.082 + 0.961 + 0.003 + {2.9}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2

$10.082 + 0.961 + 0.003 + {2.9}^{2} \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2$

Step 7.1.15

Raise

2.9

$2.9$ to the power of

2

$2$ .

10.082 + 0.961 + 0.003 + 8.41 \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2

$10.082 + 0.961 + 0.003 + 8.41 \cdot 0.2 + {(20 - (14.1))}^{2} \cdot 0.2$

Step 7.1.16

Multiply

8.41

$8.41$ by

0.2

$0.2$ .

10.082 + 0.961 + 0.003 + 1.682 + {(20 - (14.1))}^{2} \cdot 0.2

$10.082 + 0.961 + 0.003 + 1.682 + {(20 - (14.1))}^{2} \cdot 0.2$

Step 7.1.17

Multiply

- 1

$- 1$ by

14.1

$14.1$ .

10.082 + 0.961 + 0.003 + 1.682 + {(20 - 14.1)}^{2} \cdot 0.2

$10.082 + 0.961 + 0.003 + 1.682 + {(20 - 14.1)}^{2} \cdot 0.2$

Step 7.1.18

Subtract

14.1

$14.1$ from

20

$20$ .

10.082 + 0.961 + 0.003 + 1.682 + {5.9}^{2} \cdot 0.2

$10.082 + 0.961 + 0.003 + 1.682 + {5.9}^{2} \cdot 0.2$

Step 7.1.19

Raise

$5.9$ to the power of

$2$ .

$10.082 + 0.961 + 0.003 + 1.682 + 34.81 \cdot 0.2$

Step 7.1.20

Multiply

$34.81$ by

$0.2$ .

$10.082 + 0.961 + 0.003 + 1.682 + 6.962$

Step 7.2

Simplify by adding numbers.

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Step 7.2.1

Add

$10.082$ and

$0.961$ .

$11.043 + 0.003 + 1.682 + 6.962$

Step 7.2.2

Add

$11.043$ and

$0.003$ .

$11.046 + 1.682 + 6.962$

Step 7.2.3

Add

$11.046$ and

$1.682$ .

$12.728 + 6.962$

Step 7.2.4

Add

$12.728$ and

$6.962$ .

$19.69$

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