Finite Math Examples

$\begin{array}{c} x & P (x) \\ 18 & 0.2 \\ 19 & 0.1 \\ 20 & 0.3 \\ 21 & 0.2 \\ 22 & 0.2 \end{array}$

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 $x$ takes a set of separate values (such as $0$ , $1$ , $2$ ...). Its probability distribution assigns a probability $P (x)$ to each possible value $x$ . For each $x$ , the probability $P (x)$ falls between $0$ and $1$ inclusive and the sum of the probabilities for all the possible $x$ values equals to $1$ .

1. For each $x$ , $0 \leq P (x) \leq 1$ .

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

Step 1.2

$0.2$ is between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

$0.2$ is between $0$ and $1$ inclusive

Step 1.3

$0.1$ is between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

$0.1$ is between $0$ and $1$ inclusive

Step 1.4

$0.3$ is between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

$0.3$ is between $0$ and $1$ inclusive

Step 1.5

$0.2$ is between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

$0.2$ is between $0$ and $1$ inclusive

Step 1.6

For each $x$ , the probability $P (x)$ falls between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

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

Step 1.7

Find the sum of the probabilities for all the possible $x$ values.

$0.2 + 0.1 + 0.3 + 0.2 + 0.2$

Step 1.8

The sum of the probabilities for all the possible $x$ values is $0.2 + 0.1 + 0.3 + 0.2 + 0.2 = 1$ .

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Step 1.8.1

Add $0.2$ and $0.1$ .

$0.3 + 0.3 + 0.2 + 0.2$

Step 1.8.2

Add $0.3$ and $0.3$ .

$0.6 + 0.2 + 0.2$

Step 1.8.3

Add $0.6$ and $0.2$ .

$0.8 + 0.2$

Step 1.8.4

Add $0.8$ and $0.2$ .

$1$

Step 1.9

For each $x$ , the probability of $P (x)$ falls between $0$ and $1$ inclusive. In addition, the sum of the probabilities for all the possible $x$ equals $1$ , 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: $0 \leq P (x) \leq 1$ for all $x$ values

Property 2: $0.2 + 0.1 + 0.3 + 0.2 + 0.2 = 1$

The table satisfies the two properties of a probability distribution:

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

Property 2: $0.2 + 0.1 + 0.3 + 0.2 + 0.2 = 1$

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.

$u = 18 \cdot 0.2 + 19 \cdot 0.1 + 20 \cdot 0.3 + 21 \cdot 0.2 + 22 \cdot 0.2$

Step 3

Simplify each term.

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Step 3.1

Multiply $18$ by $0.2$ .

$u = 3.6 + 19 \cdot 0.1 + 20 \cdot 0.3 + 21 \cdot 0.2 + 22 \cdot 0.2$

Step 3.2

Multiply $19$ by $0.1$ .

$u = 3.6 + 1.9 + 20 \cdot 0.3 + 21 \cdot 0.2 + 22 \cdot 0.2$

Step 3.3

Multiply $20$ by $0.3$ .

$u = 3.6 + 1.9 + 6 + 21 \cdot 0.2 + 22 \cdot 0.2$

Step 3.4

Multiply $21$ by $0.2$ .

$u = 3.6 + 1.9 + 6 + 4.2 + 22 \cdot 0.2$

Step 3.5

Multiply $22$ by $0.2$ .

$u = 3.6 + 1.9 + 6 + 4.2 + 4.4$

Step 4

Simplify by adding numbers.

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Step 4.1

Add $3.6$ and $1.9$ .

$u = 5.5 + 6 + 4.2 + 4.4$

Step 4.2

Add $5.5$ and $6$ .

$u = 11.5 + 4.2 + 4.4$

Step 4.3

Add $11.5$ and $4.2$ .

$u = 15.7 + 4.4$

Step 4.4

Add $15.7$ and $4.4$ .

$u = 20.1$

Step 5

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

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

Step 6

Fill in the known values.

${(18 - (20.1))}^{2} \cdot 0.2 + {(19 - (20.1))}^{2} \cdot 0.1 + {(20 - (20.1))}^{2} \cdot 0.3 + {(21 - (20.1))}^{2} \cdot 0.2 + {(22 - (20.1))}^{2} \cdot 0.2$

Step 7

Simplify the expression.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $- 1$ by $20.1$ .

${(18 - 20.1)}^{2} \cdot 0.2 + {(19 - (20.1))}^{2} \cdot 0.1 + {(20 - (20.1))}^{2} \cdot 0.3 + {(21 - (20.1))}^{2} \cdot 0.2 + {(22 - (20.1))}^{2} \cdot 0.2$

Step 7.1.2

Subtract $20.1$ from $18$ .

${(- 2.1)}^{2} \cdot 0.2 + {(19 - (20.1))}^{2} \cdot 0.1 + {(20 - (20.1))}^{2} \cdot 0.3 + {(21 - (20.1))}^{2} \cdot 0.2 + {(22 - (20.1))}^{2} \cdot 0.2$

Step 7.1.3

Raise $- 2.1$ to the power of $2$ .

$4.41 \cdot 0.2 + {(19 - (20.1))}^{2} \cdot 0.1 + {(20 - (20.1))}^{2} \cdot 0.3 + {(21 - (20.1))}^{2} \cdot 0.2 + {(22 - (20.1))}^{2} \cdot 0.2$

Step 7.1.4

Multiply $4.41$ by $0.2$ .

$0.882 + {(19 - (20.1))}^{2} \cdot 0.1 + {(20 - (20.1))}^{2} \cdot 0.3 + {(21 - (20.1))}^{2} \cdot 0.2 + {(22 - (20.1))}^{2} \cdot 0.2$

Step 7.1.5

Multiply $- 1$ by $20.1$ .

$0.882 + {(19 - 20.1)}^{2} \cdot 0.1 + {(20 - (20.1))}^{2} \cdot 0.3 + {(21 - (20.1))}^{2} \cdot 0.2 + {(22 - (20.1))}^{2} \cdot 0.2$

Step 7.1.6

Subtract $20.1$ from $19$ .

$0.882 + {(- 1.1)}^{2} \cdot 0.1 + {(20 - (20.1))}^{2} \cdot 0.3 + {(21 - (20.1))}^{2} \cdot 0.2 + {(22 - (20.1))}^{2} \cdot 0.2$

Step 7.1.7

Raise $- 1.1$ to the power of $2$ .

$0.882 + 1.21 \cdot 0.1 + {(20 - (20.1))}^{2} \cdot 0.3 + {(21 - (20.1))}^{2} \cdot 0.2 + {(22 - (20.1))}^{2} \cdot 0.2$

Step 7.1.8

Multiply $1.21$ by $0.1$ .

$0.882 + 0.121 + {(20 - (20.1))}^{2} \cdot 0.3 + {(21 - (20.1))}^{2} \cdot 0.2 + {(22 - (20.1))}^{2} \cdot 0.2$

Step 7.1.9

Multiply $- 1$ by $20.1$ .

$0.882 + 0.121 + {(20 - 20.1)}^{2} \cdot 0.3 + {(21 - (20.1))}^{2} \cdot 0.2 + {(22 - (20.1))}^{2} \cdot 0.2$

Step 7.1.10

Subtract $20.1$ from $20$ .

$0.882 + 0.121 + {(- 0.1)}^{2} \cdot 0.3 + {(21 - (20.1))}^{2} \cdot 0.2 + {(22 - (20.1))}^{2} \cdot 0.2$

Step 7.1.11

Raise $- 0.1$ to the power of $2$ .

$0.882 + 0.121 + 0.01 \cdot 0.3 + {(21 - (20.1))}^{2} \cdot 0.2 + {(22 - (20.1))}^{2} \cdot 0.2$

Step 7.1.12

Multiply $0.01$ by $0.3$ .

$0.882 + 0.121 + 0.003 + {(21 - (20.1))}^{2} \cdot 0.2 + {(22 - (20.1))}^{2} \cdot 0.2$

Step 7.1.13

Multiply $- 1$ by $20.1$ .

$0.882 + 0.121 + 0.003 + {(21 - 20.1)}^{2} \cdot 0.2 + {(22 - (20.1))}^{2} \cdot 0.2$

Step 7.1.14

Subtract $20.1$ from $21$ .

$0.882 + 0.121 + 0.003 + {0.9}^{2} \cdot 0.2 + {(22 - (20.1))}^{2} \cdot 0.2$

Step 7.1.15

Raise $0.9$ to the power of $2$ .

$0.882 + 0.121 + 0.003 + 0.81 \cdot 0.2 + {(22 - (20.1))}^{2} \cdot 0.2$

Step 7.1.16

Multiply $0.81$ by $0.2$ .

$0.882 + 0.121 + 0.003 + 0.162 + {(22 - (20.1))}^{2} \cdot 0.2$

Step 7.1.17

Multiply $- 1$ by $20.1$ .

$0.882 + 0.121 + 0.003 + 0.162 + {(22 - 20.1)}^{2} \cdot 0.2$

Step 7.1.18

Subtract $20.1$ from $22$ .

$0.882 + 0.121 + 0.003 + 0.162 + {1.9}^{2} \cdot 0.2$

Step 7.1.19

Raise $1.9$ to the power of $2$ .

$0.882 + 0.121 + 0.003 + 0.162 + 3.61 \cdot 0.2$

Step 7.1.20

Multiply $3.61$ by $0.2$ .

$0.882 + 0.121 + 0.003 + 0.162 + 0.722$

Step 7.2

Simplify by adding numbers.

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Step 7.2.1

Add $0.882$ and $0.121$ .

$1.003 + 0.003 + 0.162 + 0.722$

Step 7.2.2

Add $1.003$ and $0.003$ .

$1.006 + 0.162 + 0.722$

Step 7.2.3

Add $1.006$ and $0.162$ .

$1.168 + 0.722$

Step 7.2.4

Add $1.168$ and $0.722$ .

$1.89$