Finite Math Examples

\begin{array}{c} x & P (x) \\ 6 & 0.1 \\ 9 & 0.2 \\ 13 & 0.3 \\ 16 & 0.4 \end{array}

$\begin{array}{c} x & P (x) \\ 6 & 0.1 \\ 9 & 0.2 \\ 13 & 0.3 \\ 16 & 0.4 \end{array}$

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

0.4

$0.4$ is between

0

$0$ and

1

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

0.4

$0.4$ 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 + 0.4

$0.1 + 0.2 + 0.3 + 0.4$

Step 1.8

The sum of the probabilities for all the possible

x

$x$ values is

0.1 + 0.2 + 0.3 + 0.4 = 1

$0.1 + 0.2 + 0.3 + 0.4 = 1$ .

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Step 1.8.1

Add

0.1

$0.1$ and

0.2

$0.2$ .

0.3 + 0.3 + 0.4

$0.3 + 0.3 + 0.4$

Step 1.8.2

Add

0.3

$0.3$ and

0.3

$0.3$ .

0.6 + 0.4

$0.6 + 0.4$

Step 1.8.3

Add

0.6

$0.6$ and

0.4

$0.4$ .

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 + 0.4 = 1

$0.1 + 0.2 + 0.3 + 0.4 = 1$

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 + 0.4 = 1

$0.1 + 0.2 + 0.3 + 0.4 = 1$

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 = 6 \cdot 0.1 + 9 \cdot 0.2 + 13 \cdot 0.3 + 16 \cdot 0.4

$u = 6 \cdot 0.1 + 9 \cdot 0.2 + 13 \cdot 0.3 + 16 \cdot 0.4$

Step 3

Simplify each term.

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Step 3.1

Multiply

6

$6$ by

0.1

$0.1$ .

u = 0.6 + 9 \cdot 0.2 + 13 \cdot 0.3 + 16 \cdot 0.4

$u = 0.6 + 9 \cdot 0.2 + 13 \cdot 0.3 + 16 \cdot 0.4$

Step 3.2

Multiply

9

$9$ by

0.2

$0.2$ .

u = 0.6 + 1.8 + 13 \cdot 0.3 + 16 \cdot 0.4

$u = 0.6 + 1.8 + 13 \cdot 0.3 + 16 \cdot 0.4$

Step 3.3

Multiply

13

$13$ by

0.3

$0.3$ .

u = 0.6 + 1.8 + 3.9 + 16 \cdot 0.4

$u = 0.6 + 1.8 + 3.9 + 16 \cdot 0.4$

Step 3.4

Multiply

16

$16$ by

0.4

$0.4$ .

u = 0.6 + 1.8 + 3.9 + 6.4

$u = 0.6 + 1.8 + 3.9 + 6.4$

u = 0.6 + 1.8 + 3.9 + 6.4

$u = 0.6 + 1.8 + 3.9 + 6.4$

Step 4

Simplify by adding numbers.

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Step 4.1

Add

0.6

$0.6$ and

1.8

$1.8$ .

u = 2.4 + 3.9 + 6.4

$u = 2.4 + 3.9 + 6.4$

Step 4.2

Add

2.4

$2.4$ and

3.9

$3.9$ .

u = 6.3 + 6.4

$u = 6.3 + 6.4$

Step 4.3

Add

6.3

$6.3$ and

6.4

$6.4$ .

u = 12.7

$u = 12.7$

u = 12.7

$u = 12.7$

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.

{(6 - (12.7))}^{2} \cdot 0.1 + {(9 - (12.7))}^{2} \cdot 0.2 + {(13 - (12.7))}^{2} \cdot 0.3 + {(16 - (12.7))}^{2} \cdot 0.4

${(6 - (12.7))}^{2} \cdot 0.1 + {(9 - (12.7))}^{2} \cdot 0.2 + {(13 - (12.7))}^{2} \cdot 0.3 + {(16 - (12.7))}^{2} \cdot 0.4$

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

12.7

$12.7$ .

{(6 - 12.7)}^{2} \cdot 0.1 + {(9 - (12.7))}^{2} \cdot 0.2 + {(13 - (12.7))}^{2} \cdot 0.3 + {(16 - (12.7))}^{2} \cdot 0.4

${(6 - 12.7)}^{2} \cdot 0.1 + {(9 - (12.7))}^{2} \cdot 0.2 + {(13 - (12.7))}^{2} \cdot 0.3 + {(16 - (12.7))}^{2} \cdot 0.4$

Step 7.1.2

Subtract

12.7

$12.7$ from

6

$6$ .

{(- 6.7)}^{2} \cdot 0.1 + {(9 - (12.7))}^{2} \cdot 0.2 + {(13 - (12.7))}^{2} \cdot 0.3 + {(16 - (12.7))}^{2} \cdot 0.4

${(- 6.7)}^{2} \cdot 0.1 + {(9 - (12.7))}^{2} \cdot 0.2 + {(13 - (12.7))}^{2} \cdot 0.3 + {(16 - (12.7))}^{2} \cdot 0.4$

Step 7.1.3

Raise

- 6.7

$- 6.7$ to the power of

2

$2$ .

44.89 \cdot 0.1 + {(9 - (12.7))}^{2} \cdot 0.2 + {(13 - (12.7))}^{2} \cdot 0.3 + {(16 - (12.7))}^{2} \cdot 0.4

$44.89 \cdot 0.1 + {(9 - (12.7))}^{2} \cdot 0.2 + {(13 - (12.7))}^{2} \cdot 0.3 + {(16 - (12.7))}^{2} \cdot 0.4$

Step 7.1.4

Multiply

44.89

$44.89$ by

0.1

$0.1$ .

4.489 + {(9 - (12.7))}^{2} \cdot 0.2 + {(13 - (12.7))}^{2} \cdot 0.3 + {(16 - (12.7))}^{2} \cdot 0.4

$4.489 + {(9 - (12.7))}^{2} \cdot 0.2 + {(13 - (12.7))}^{2} \cdot 0.3 + {(16 - (12.7))}^{2} \cdot 0.4$

Step 7.1.5

Multiply

- 1

$- 1$ by

12.7

$12.7$ .

4.489 + {(9 - 12.7)}^{2} \cdot 0.2 + {(13 - (12.7))}^{2} \cdot 0.3 + {(16 - (12.7))}^{2} \cdot 0.4

$4.489 + {(9 - 12.7)}^{2} \cdot 0.2 + {(13 - (12.7))}^{2} \cdot 0.3 + {(16 - (12.7))}^{2} \cdot 0.4$

Step 7.1.6

Subtract

12.7

$12.7$ from

9

$9$ .

4.489 + {(- 3.7)}^{2} \cdot 0.2 + {(13 - (12.7))}^{2} \cdot 0.3 + {(16 - (12.7))}^{2} \cdot 0.4

$4.489 + {(- 3.7)}^{2} \cdot 0.2 + {(13 - (12.7))}^{2} \cdot 0.3 + {(16 - (12.7))}^{2} \cdot 0.4$

Step 7.1.7

Raise

- 3.7

$- 3.7$ to the power of

2

$2$ .

4.489 + 13.69 \cdot 0.2 + {(13 - (12.7))}^{2} \cdot 0.3 + {(16 - (12.7))}^{2} \cdot 0.4

$4.489 + 13.69 \cdot 0.2 + {(13 - (12.7))}^{2} \cdot 0.3 + {(16 - (12.7))}^{2} \cdot 0.4$

Step 7.1.8

Multiply

13.69

$13.69$ by

0.2

$0.2$ .

4.489 + 2.738 + {(13 - (12.7))}^{2} \cdot 0.3 + {(16 - (12.7))}^{2} \cdot 0.4

$4.489 + 2.738 + {(13 - (12.7))}^{2} \cdot 0.3 + {(16 - (12.7))}^{2} \cdot 0.4$

Step 7.1.9

Multiply

- 1

$- 1$ by

12.7

$12.7$ .

4.489 + 2.738 + {(13 - 12.7)}^{2} \cdot 0.3 + {(16 - (12.7))}^{2} \cdot 0.4

$4.489 + 2.738 + {(13 - 12.7)}^{2} \cdot 0.3 + {(16 - (12.7))}^{2} \cdot 0.4$

Step 7.1.10

Subtract

12.7

$12.7$ from

13

$13$ .

4.489 + 2.738 + {0.3}^{2} \cdot 0.3 + {(16 - (12.7))}^{2} \cdot 0.4

$4.489 + 2.738 + {0.3}^{2} \cdot 0.3 + {(16 - (12.7))}^{2} \cdot 0.4$

Step 7.1.11

Multiply

{0.3}^{2}

${0.3}^{2}$ by

0.3

$0.3$ by adding the exponents.

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Step 7.1.11.1

Multiply

{0.3}^{2}

${0.3}^{2}$ by

0.3

$0.3$ .

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Step 7.1.11.1.1

Raise

0.3

$0.3$ to the power of

1

$1$ .

4.489 + 2.738 + {0.3}^{2} \cdot {0.3}^{1} + {(16 - (12.7))}^{2} \cdot 0.4

$4.489 + 2.738 + {0.3}^{2} \cdot {0.3}^{1} + {(16 - (12.7))}^{2} \cdot 0.4$

Step 7.1.11.1.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

4.489 + 2.738 + {0.3}^{2 + 1} + {(16 - (12.7))}^{2} \cdot 0.4

$4.489 + 2.738 + {0.3}^{2 + 1} + {(16 - (12.7))}^{2} \cdot 0.4$

4.489 + 2.738 + {0.3}^{2 + 1} + {(16 - (12.7))}^{2} \cdot 0.4

$4.489 + 2.738 + {0.3}^{2 + 1} + {(16 - (12.7))}^{2} \cdot 0.4$

Step 7.1.11.2

Add

2

$2$ and

1

$1$ .

4.489 + 2.738 + {0.3}^{3} + {(16 - (12.7))}^{2} \cdot 0.4

$4.489 + 2.738 + {0.3}^{3} + {(16 - (12.7))}^{2} \cdot 0.4$

4.489 + 2.738 + {0.3}^{3} + {(16 - (12.7))}^{2} \cdot 0.4

$4.489 + 2.738 + {0.3}^{3} + {(16 - (12.7))}^{2} \cdot 0.4$

Step 7.1.12

Raise

0.3

$0.3$ to the power of

3

$3$ .

4.489 + 2.738 + 0.027 + {(16 - (12.7))}^{2} \cdot 0.4

$4.489 + 2.738 + 0.027 + {(16 - (12.7))}^{2} \cdot 0.4$

Step 7.1.13

Multiply

- 1

$- 1$ by

12.7

$12.7$ .

4.489 + 2.738 + 0.027 + {(16 - 12.7)}^{2} \cdot 0.4

$4.489 + 2.738 + 0.027 + {(16 - 12.7)}^{2} \cdot 0.4$

Step 7.1.14

Subtract

12.7

$12.7$ from

16

$16$ .

4.489 + 2.738 + 0.027 + {3.3}^{2} \cdot 0.4

$4.489 + 2.738 + 0.027 + {3.3}^{2} \cdot 0.4$

Step 7.1.15

Raise

3.3

$3.3$ to the power of

2

$2$ .

4.489 + 2.738 + 0.027 + 10.89 \cdot 0.4

$4.489 + 2.738 + 0.027 + 10.89 \cdot 0.4$

Step 7.1.16

Multiply

10.89

$10.89$ by

0.4

$0.4$ .

4.489 + 2.738 + 0.027 + 4.356

$4.489 + 2.738 + 0.027 + 4.356$

4.489 + 2.738 + 0.027 + 4.356

$4.489 + 2.738 + 0.027 + 4.356$

Step 7.2

Simplify by adding numbers.

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Step 7.2.1

Add

4.489

$4.489$ and

2.738

$2.738$ .

7.227 + 0.027 + 4.356

$7.227 + 0.027 + 4.356$

Step 7.2.2

Add

7.227

$7.227$ and

0.027

$0.027$ .

7.254 + 4.356

$7.254 + 4.356$

Step 7.2.3

Add

7.254

$7.254$ and

4.356

$4.356$ .

11.61

$11.61$

11.61

$11.61$

11.61

$11.61$

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