Statistics Examples

$\begin{array}{c} x & P (x) \\ 2 & 0.3 \\ 5 & 0.1 \\ 8 & 0.1 \\ 11 & 0.1 \\ 14 & 0.4 \end{array}$

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 $x$ 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

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

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

Step 1.3

$0.1$ 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

$0.4$ is between $0$ and $1$ inclusive, which meets the first property of the probability distribution.

$0.4$ is between $0$ and $1$ inclusive

Step 1.5

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 1.6

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

$0.3 + 0.1 + 0.1 + 0.1 + 0.4$

Step 1.7

The sum of the probabilities for all the possible $x$ values is $0.3 + 0.1 + 0.1 + 0.1 + 0.4 = 1$ .

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Step 1.7.1

Add $0.3$ and $0.1$ .

$0.4 + 0.1 + 0.1 + 0.4$

Step 1.7.2

Add $0.4$ and $0.1$ .

$0.5 + 0.1 + 0.4$

Step 1.7.3

Add $0.5$ and $0.1$ .

$0.6 + 0.4$

Step 1.7.4

Add $0.6$ and $0.4$ .

$1$

Step 1.8

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: $0.3 + 0.1 + 0.1 + 0.1 + 0.4 = 1$

The table satisfies the two properties of a probability distribution:

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

Property 2: $0.3 + 0.1 + 0.1 + 0.1 + 0.4 = 1$

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.

$u = 2 \cdot 0.3 + 5 \cdot 0.1 + 8 \cdot 0.1 + 11 \cdot 0.1 + 14 \cdot 0.4$

Step 3

Simplify each term.

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Step 3.1

Multiply $2$ by $0.3$ .

$u = 0.6 + 5 \cdot 0.1 + 8 \cdot 0.1 + 11 \cdot 0.1 + 14 \cdot 0.4$

Step 3.2

Multiply $5$ by $0.1$ .

$u = 0.6 + 0.5 + 8 \cdot 0.1 + 11 \cdot 0.1 + 14 \cdot 0.4$

Step 3.3

Multiply $8$ by $0.1$ .

$u = 0.6 + 0.5 + 0.8 + 11 \cdot 0.1 + 14 \cdot 0.4$

Step 3.4

Multiply $11$ by $0.1$ .

$u = 0.6 + 0.5 + 0.8 + 1.1 + 14 \cdot 0.4$

Step 3.5

Multiply $14$ by $0.4$ .

$u = 0.6 + 0.5 + 0.8 + 1.1 + 5.6$

Step 4

Simplify by adding numbers.

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Step 4.1

Add $0.6$ and $0.5$ .

$u = 1.1 + 0.8 + 1.1 + 5.6$

Step 4.2

Add $1.1$ and $0.8$ .

$u = 1.9 + 1.1 + 5.6$

Step 4.3

Add $1.9$ and $1.1$ .

$u = 3 + 5.6$

Step 4.4

Add $3$ and $5.6$ .

$u = 8.6$

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.

${(2 - (8.6))}^{2} \cdot 0.3 + {(5 - (8.6))}^{2} \cdot 0.1 + {(8 - (8.6))}^{2} \cdot 0.1 + {(11 - (8.6))}^{2} \cdot 0.1 + {(14 - (8.6))}^{2} \cdot 0.4$

Step 7

Simplify the expression.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $- 1$ by $8.6$ .

${(2 - 8.6)}^{2} \cdot 0.3 + {(5 - (8.6))}^{2} \cdot 0.1 + {(8 - (8.6))}^{2} \cdot 0.1 + {(11 - (8.6))}^{2} \cdot 0.1 + {(14 - (8.6))}^{2} \cdot 0.4$

Step 7.1.2

Subtract $8.6$ from $2$ .

${(- 6.6)}^{2} \cdot 0.3 + {(5 - (8.6))}^{2} \cdot 0.1 + {(8 - (8.6))}^{2} \cdot 0.1 + {(11 - (8.6))}^{2} \cdot 0.1 + {(14 - (8.6))}^{2} \cdot 0.4$

Step 7.1.3

Raise $- 6.6$ to the power of $2$ .

$43.56 \cdot 0.3 + {(5 - (8.6))}^{2} \cdot 0.1 + {(8 - (8.6))}^{2} \cdot 0.1 + {(11 - (8.6))}^{2} \cdot 0.1 + {(14 - (8.6))}^{2} \cdot 0.4$

Step 7.1.4

Multiply $43.56$ by $0.3$ .

$13.068 + {(5 - (8.6))}^{2} \cdot 0.1 + {(8 - (8.6))}^{2} \cdot 0.1 + {(11 - (8.6))}^{2} \cdot 0.1 + {(14 - (8.6))}^{2} \cdot 0.4$

Step 7.1.5

Multiply $- 1$ by $8.6$ .

$13.068 + {(5 - 8.6)}^{2} \cdot 0.1 + {(8 - (8.6))}^{2} \cdot 0.1 + {(11 - (8.6))}^{2} \cdot 0.1 + {(14 - (8.6))}^{2} \cdot 0.4$

Step 7.1.6

Subtract $8.6$ from $5$ .

$13.068 + {(- 3.6)}^{2} \cdot 0.1 + {(8 - (8.6))}^{2} \cdot 0.1 + {(11 - (8.6))}^{2} \cdot 0.1 + {(14 - (8.6))}^{2} \cdot 0.4$

Step 7.1.7

Raise $- 3.6$ to the power of $2$ .

$13.068 + 12.96 \cdot 0.1 + {(8 - (8.6))}^{2} \cdot 0.1 + {(11 - (8.6))}^{2} \cdot 0.1 + {(14 - (8.6))}^{2} \cdot 0.4$

Step 7.1.8

Multiply $12.96$ by $0.1$ .

$13.068 + 1.296 + {(8 - (8.6))}^{2} \cdot 0.1 + {(11 - (8.6))}^{2} \cdot 0.1 + {(14 - (8.6))}^{2} \cdot 0.4$

Step 7.1.9

Multiply $- 1$ by $8.6$ .

$13.068 + 1.296 + {(8 - 8.6)}^{2} \cdot 0.1 + {(11 - (8.6))}^{2} \cdot 0.1 + {(14 - (8.6))}^{2} \cdot 0.4$

Step 7.1.10

Subtract $8.6$ from $8$ .

$13.068 + 1.296 + {(- 0.6)}^{2} \cdot 0.1 + {(11 - (8.6))}^{2} \cdot 0.1 + {(14 - (8.6))}^{2} \cdot 0.4$

Step 7.1.11

Raise $- 0.6$ to the power of $2$ .

$13.068 + 1.296 + 0.36 \cdot 0.1 + {(11 - (8.6))}^{2} \cdot 0.1 + {(14 - (8.6))}^{2} \cdot 0.4$

Step 7.1.12

Multiply $0.36$ by $0.1$ .

$13.068 + 1.296 + 0.036 + {(11 - (8.6))}^{2} \cdot 0.1 + {(14 - (8.6))}^{2} \cdot 0.4$

Step 7.1.13

Multiply $- 1$ by $8.6$ .

$13.068 + 1.296 + 0.036 + {(11 - 8.6)}^{2} \cdot 0.1 + {(14 - (8.6))}^{2} \cdot 0.4$

Step 7.1.14

Subtract $8.6$ from $11$ .

$13.068 + 1.296 + 0.036 + {2.4}^{2} \cdot 0.1 + {(14 - (8.6))}^{2} \cdot 0.4$

Step 7.1.15

Raise $2.4$ to the power of $2$ .

$13.068 + 1.296 + 0.036 + 5.76 \cdot 0.1 + {(14 - (8.6))}^{2} \cdot 0.4$

Step 7.1.16

Multiply $5.76$ by $0.1$ .

$13.068 + 1.296 + 0.036 + 0.576 + {(14 - (8.6))}^{2} \cdot 0.4$

Step 7.1.17

Multiply $- 1$ by $8.6$ .

$13.068 + 1.296 + 0.036 + 0.576 + {(14 - 8.6)}^{2} \cdot 0.4$

Step 7.1.18

Subtract $8.6$ from $14$ .

$13.068 + 1.296 + 0.036 + 0.576 + {5.4}^{2} \cdot 0.4$

Step 7.1.19

Raise $5.4$ to the power of $2$ .

$13.068 + 1.296 + 0.036 + 0.576 + 29.16 \cdot 0.4$

Step 7.1.20

Multiply $29.16$ by $0.4$ .

$13.068 + 1.296 + 0.036 + 0.576 + 11.664$

Step 7.2

Simplify by adding numbers.

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Step 7.2.1

Add $13.068$ and $1.296$ .

$14.364 + 0.036 + 0.576 + 11.664$

Step 7.2.2

Add $14.364$ and $0.036$ .

$14.4 + 0.576 + 11.664$

Step 7.2.3

Add $14.4$ and $0.576$ .

$14.976 + 11.664$

Step 7.2.4

Add $14.976$ and $11.664$ .

$26.64$

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