Finite Math Examples

\begin{array}{c} x & P (x) \\ 1 & 0.4 \\ 5 & 0.1 \\ 8 & 0.2 \\ 1 & 0.1 \\ 14 & 0.2 \end{array}

$\begin{array}{c} x & P (x) \\ 1 & 0.4 \\ 5 & 0.1 \\ 8 & 0.2 \\ 1 & 0.1 \\ 14 & 0.2 \end{array}$

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

0.4

$0.4$ 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 1.3

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

0.2

$0.2$ 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 1.5

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 1.6

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 1.7

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 1.8

Find the sum of the probabilities for all the possible

x

$x$ values.

0.4 + 0.1 + 0.2 + 0.1 + 0.2

$0.4 + 0.1 + 0.2 + 0.1 + 0.2$

Step 1.9

The sum of the probabilities for all the possible

x

$x$ values is

0.4 + 0.1 + 0.2 + 0.1 + 0.2 = 1

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

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Step 1.9.1

Add

0.4

$0.4$ and

0.1

$0.1$ .

0.5 + 0.2 + 0.1 + 0.2

$0.5 + 0.2 + 0.1 + 0.2$

Step 1.9.2

Add

0.5

$0.5$ and

0.2

$0.2$ .

0.7 + 0.1 + 0.2

$0.7 + 0.1 + 0.2$

Step 1.9.3

Add

0.7

$0.7$ and

0.1

$0.1$ .

0.8 + 0.2

$0.8 + 0.2$

Step 1.9.4

Add

0.8

$0.8$ and

0.2

$0.2$ .

1

$1$

1

$1$

Step 1.10

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:

0.4 + 0.1 + 0.2 + 0.1 + 0.2 = 1

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

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:

0.4 + 0.1 + 0.2 + 0.1 + 0.2 = 1

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

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.

u = 1 \cdot 0.4 + 5 \cdot 0.1 + 8 \cdot 0.2 + 1 \cdot 0.1 + 14 \cdot 0.2

$u = 1 \cdot 0.4 + 5 \cdot 0.1 + 8 \cdot 0.2 + 1 \cdot 0.1 + 14 \cdot 0.2$

Step 3

Simplify each term.

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Step 3.1

Multiply

0.4

$0.4$ by

1

$1$ .

u = 0.4 + 5 \cdot 0.1 + 8 \cdot 0.2 + 1 \cdot 0.1 + 14 \cdot 0.2

$u = 0.4 + 5 \cdot 0.1 + 8 \cdot 0.2 + 1 \cdot 0.1 + 14 \cdot 0.2$

Step 3.2

Multiply

5

$5$ by

0.1

$0.1$ .

u = 0.4 + 0.5 + 8 \cdot 0.2 + 1 \cdot 0.1 + 14 \cdot 0.2

$u = 0.4 + 0.5 + 8 \cdot 0.2 + 1 \cdot 0.1 + 14 \cdot 0.2$

Step 3.3

Multiply

8

$8$ by

0.2

$0.2$ .

u = 0.4 + 0.5 + 1.6 + 1 \cdot 0.1 + 14 \cdot 0.2

$u = 0.4 + 0.5 + 1.6 + 1 \cdot 0.1 + 14 \cdot 0.2$

Step 3.4

Multiply

0.1

$0.1$ by

1

$1$ .

u = 0.4 + 0.5 + 1.6 + 0.1 + 14 \cdot 0.2

$u = 0.4 + 0.5 + 1.6 + 0.1 + 14 \cdot 0.2$

Step 3.5

Multiply

14

$14$ by

0.2

$0.2$ .

u = 0.4 + 0.5 + 1.6 + 0.1 + 2.8

$u = 0.4 + 0.5 + 1.6 + 0.1 + 2.8$

u = 0.4 + 0.5 + 1.6 + 0.1 + 2.8

$u = 0.4 + 0.5 + 1.6 + 0.1 + 2.8$

Step 4

Simplify by adding numbers.

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Step 4.1

Add

0.4

$0.4$ and

0.5

$0.5$ .

u = 0.9 + 1.6 + 0.1 + 2.8

$u = 0.9 + 1.6 + 0.1 + 2.8$

Step 4.2

Add

0.9

$0.9$ and

1.6

$1.6$ .

u = 2.5 + 0.1 + 2.8

$u = 2.5 + 0.1 + 2.8$

Step 4.3

Add

2.5

$2.5$ and

0.1

$0.1$ .

u = 2.6 + 2.8

$u = 2.6 + 2.8$

Step 4.4

Add

2.6

$2.6$ and

2.8

$2.8$ .

u = 5.4

$u = 5.4$

u = 5.4

$u = 5.4$

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.

{(1 - (5.4))}^{2} \cdot 0.4 + {(5 - (5.4))}^{2} \cdot 0.1 + {(8 - (5.4))}^{2} \cdot 0.2 + {(1 - (5.4))}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2

${(1 - (5.4))}^{2} \cdot 0.4 + {(5 - (5.4))}^{2} \cdot 0.1 + {(8 - (5.4))}^{2} \cdot 0.2 + {(1 - (5.4))}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2$

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

5.4

$5.4$ .

{(1 - 5.4)}^{2} \cdot 0.4 + {(5 - (5.4))}^{2} \cdot 0.1 + {(8 - (5.4))}^{2} \cdot 0.2 + {(1 - (5.4))}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2

${(1 - 5.4)}^{2} \cdot 0.4 + {(5 - (5.4))}^{2} \cdot 0.1 + {(8 - (5.4))}^{2} \cdot 0.2 + {(1 - (5.4))}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2$

Step 7.1.2

Subtract

5.4

$5.4$ from

1

$1$ .

{(- 4.4)}^{2} \cdot 0.4 + {(5 - (5.4))}^{2} \cdot 0.1 + {(8 - (5.4))}^{2} \cdot 0.2 + {(1 - (5.4))}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2

${(- 4.4)}^{2} \cdot 0.4 + {(5 - (5.4))}^{2} \cdot 0.1 + {(8 - (5.4))}^{2} \cdot 0.2 + {(1 - (5.4))}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2$

Step 7.1.3

Raise

- 4.4

$- 4.4$ to the power of

2

$2$ .

19.36 \cdot 0.4 + {(5 - (5.4))}^{2} \cdot 0.1 + {(8 - (5.4))}^{2} \cdot 0.2 + {(1 - (5.4))}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2

$19.36 \cdot 0.4 + {(5 - (5.4))}^{2} \cdot 0.1 + {(8 - (5.4))}^{2} \cdot 0.2 + {(1 - (5.4))}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2$

Step 7.1.4

Multiply

19.36

$19.36$ by

0.4

$0.4$ .

7.744 + {(5 - (5.4))}^{2} \cdot 0.1 + {(8 - (5.4))}^{2} \cdot 0.2 + {(1 - (5.4))}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2

$7.744 + {(5 - (5.4))}^{2} \cdot 0.1 + {(8 - (5.4))}^{2} \cdot 0.2 + {(1 - (5.4))}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2$

Step 7.1.5

Multiply

- 1

$- 1$ by

5.4

$5.4$ .

7.744 + {(5 - 5.4)}^{2} \cdot 0.1 + {(8 - (5.4))}^{2} \cdot 0.2 + {(1 - (5.4))}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2

$7.744 + {(5 - 5.4)}^{2} \cdot 0.1 + {(8 - (5.4))}^{2} \cdot 0.2 + {(1 - (5.4))}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2$

Step 7.1.6

Subtract

5.4

$5.4$ from

5

$5$ .

7.744 + {(- 0.4)}^{2} \cdot 0.1 + {(8 - (5.4))}^{2} \cdot 0.2 + {(1 - (5.4))}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2

$7.744 + {(- 0.4)}^{2} \cdot 0.1 + {(8 - (5.4))}^{2} \cdot 0.2 + {(1 - (5.4))}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2$

Step 7.1.7

Raise

- 0.4

$- 0.4$ to the power of

2

$2$ .

7.744 + 0.16 \cdot 0.1 + {(8 - (5.4))}^{2} \cdot 0.2 + {(1 - (5.4))}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2

$7.744 + 0.16 \cdot 0.1 + {(8 - (5.4))}^{2} \cdot 0.2 + {(1 - (5.4))}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2$

Step 7.1.8

Multiply

0.16

$0.16$ by

0.1

$0.1$ .

7.744 + 0.016 + {(8 - (5.4))}^{2} \cdot 0.2 + {(1 - (5.4))}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2

$7.744 + 0.016 + {(8 - (5.4))}^{2} \cdot 0.2 + {(1 - (5.4))}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2$

Step 7.1.9

Multiply

- 1

$- 1$ by

5.4

$5.4$ .

7.744 + 0.016 + {(8 - 5.4)}^{2} \cdot 0.2 + {(1 - (5.4))}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2

$7.744 + 0.016 + {(8 - 5.4)}^{2} \cdot 0.2 + {(1 - (5.4))}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2$

Step 7.1.10

Subtract

5.4

$5.4$ from

8

$8$ .

7.744 + 0.016 + {2.6}^{2} \cdot 0.2 + {(1 - (5.4))}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2

$7.744 + 0.016 + {2.6}^{2} \cdot 0.2 + {(1 - (5.4))}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2$

Step 7.1.11

Raise

2.6

$2.6$ to the power of

2

$2$ .

7.744 + 0.016 + 6.76 \cdot 0.2 + {(1 - (5.4))}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2

$7.744 + 0.016 + 6.76 \cdot 0.2 + {(1 - (5.4))}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2$

Step 7.1.12

Multiply

6.76

$6.76$ by

0.2

$0.2$ .

7.744 + 0.016 + 1.352 + {(1 - (5.4))}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2

$7.744 + 0.016 + 1.352 + {(1 - (5.4))}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2$

Step 7.1.13

Multiply

- 1

$- 1$ by

5.4

$5.4$ .

7.744 + 0.016 + 1.352 + {(1 - 5.4)}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2

$7.744 + 0.016 + 1.352 + {(1 - 5.4)}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2$

Step 7.1.14

Subtract

5.4

$5.4$ from

1

$1$ .

7.744 + 0.016 + 1.352 + {(- 4.4)}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2

$7.744 + 0.016 + 1.352 + {(- 4.4)}^{2} \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2$

Step 7.1.15

Raise

- 4.4

$- 4.4$ to the power of

2

$2$ .

7.744 + 0.016 + 1.352 + 19.36 \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2

$7.744 + 0.016 + 1.352 + 19.36 \cdot 0.1 + {(14 - (5.4))}^{2} \cdot 0.2$

Step 7.1.16

Multiply

19.36

$19.36$ by

0.1

$0.1$ .

7.744 + 0.016 + 1.352 + 1.936 + {(14 - (5.4))}^{2} \cdot 0.2

$7.744 + 0.016 + 1.352 + 1.936 + {(14 - (5.4))}^{2} \cdot 0.2$

Step 7.1.17

Multiply

- 1

$- 1$ by

5.4

$5.4$ .

7.744 + 0.016 + 1.352 + 1.936 + {(14 - 5.4)}^{2} \cdot 0.2

$7.744 + 0.016 + 1.352 + 1.936 + {(14 - 5.4)}^{2} \cdot 0.2$

Step 7.1.18

Subtract

5.4

$5.4$ from

14

$14$ .

7.744 + 0.016 + 1.352 + 1.936 + {8.6}^{2} \cdot 0.2

$7.744 + 0.016 + 1.352 + 1.936 + {8.6}^{2} \cdot 0.2$

Step 7.1.19

Raise

8.6

$8.6$ to the power of

2

$2$ .

7.744 + 0.016 + 1.352 + 1.936 + 73.96 \cdot 0.2

$7.744 + 0.016 + 1.352 + 1.936 + 73.96 \cdot 0.2$

Step 7.1.20

Multiply

73.96

$73.96$ by

0.2

$0.2$ .

7.744 + 0.016 + 1.352 + 1.936 + 14.792

$7.744 + 0.016 + 1.352 + 1.936 + 14.792$

7.744 + 0.016 + 1.352 + 1.936 + 14.792

$7.744 + 0.016 + 1.352 + 1.936 + 14.792$

Step 7.2

Simplify by adding numbers.

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Step 7.2.1

Add

7.744

$7.744$ and

0.016

$0.016$ .

7.76 + 1.352 + 1.936 + 14.792

$7.76 + 1.352 + 1.936 + 14.792$

Step 7.2.2

Add

7.76

$7.76$ and

1.352

$1.352$ .

9.112 + 1.936 + 14.792

$9.112 + 1.936 + 14.792$

Step 7.2.3

Add

9.112

$9.112$ and

1.936

$1.936$ .

11.048 + 14.792

$11.048 + 14.792$

Step 7.2.4

Add

11.048

$11.048$ and

14.792

$14.792$ .

25.84

$25.84$

25.84

$25.84$

25.84

$25.84$

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