Finite Math Examples

$\begin{array}{c} x & P (x) \\ 1 & 0.29 \\ 2 & 0.45 \\ 3 & 0.12 \\ 4 & 0.14 \end{array}$

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

$x$ ,

$0 \leq P (x) \leq 1$ .

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

Step 1.2

$0.29$ is between

$0$ and

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

$0.29$ is between

$0$ and

$1$ inclusive

Step 1.3

$0.45$ is between

$0$ and

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

$0.45$ is between

$0$ and

$1$ inclusive

Step 1.4

$0.12$ is between

$0$ and

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

$0.12$ is between

$0$ and

$1$ inclusive

Step 1.5

$0.14$ is between

$0$ and

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

$0.14$ 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.

$0.29 + 0.45 + 0.12 + 0.14$

Step 1.8

The sum of the probabilities for all the possible

$x$ values is

$0.29 + 0.45 + 0.12 + 0.14 = 1$ .

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Step 1.8.1

Add

$0.29$ and

$0.45$ .

$0.74 + 0.12 + 0.14$

Step 1.8.2

Add

$0.74$ and

$0.12$ .

$0.86 + 0.14$

Step 1.8.3

Add

$0.86$ and

$0.14$ .

$1$

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:

$0.29 + 0.45 + 0.12 + 0.14 = 1$

The table satisfies the two properties of a probability distribution:

Property 1:

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

$x$ values

Property 2:

$0.29 + 0.45 + 0.12 + 0.14 = 1$

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.

$1 \cdot 0.29 + 2 \cdot 0.45 + 3 \cdot 0.12 + 4 \cdot 0.14$

Step 3

Simplify each term.

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Step 3.1

Multiply

$0.29$ by

$1$ .

$0.29 + 2 \cdot 0.45 + 3 \cdot 0.12 + 4 \cdot 0.14$

Step 3.2

Multiply

$2$ by

$0.45$ .

$0.29 + 0.9 + 3 \cdot 0.12 + 4 \cdot 0.14$

Step 3.3

Multiply

$3$ by

$0.12$ .

$0.29 + 0.9 + 0.36 + 4 \cdot 0.14$

Step 3.4

Multiply

$4$ by

$0.14$ .

$0.29 + 0.9 + 0.36 + 0.56$

Step 4

Simplify by adding numbers.

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Step 4.1

Add

$0.29$ and

$0.9$ .

$1.19 + 0.36 + 0.56$

Step 4.2

Add

$1.19$ and

$0.36$ .

$1.55 + 0.56$

Step 4.3

Add

$1.55$ and

$0.56$ .

$2.11$

Step 5

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

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

Step 6

Fill in the known values.

$\sqrt{{(1 - (2.11))}^{2} \cdot 0.29 + {(2 - (2.11))}^{2} \cdot 0.45 + {(3 - (2.11))}^{2} \cdot 0.12 + {(4 - (2.11))}^{2} \cdot 0.14}$

Step 7

Simplify the expression.

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Step 7.1

Multiply

$- 1$ by

$2.11$ .

$\sqrt{{(1 - 2.11)}^{2} \cdot 0.29 + {(2 - (2.11))}^{2} \cdot 0.45 + {(3 - (2.11))}^{2} \cdot 0.12 + {(4 - (2.11))}^{2} \cdot 0.14}$

Step 7.2

Subtract

$2.11$ from

$1$ .

$\sqrt{{(- 1.11)}^{2} \cdot 0.29 + {(2 - (2.11))}^{2} \cdot 0.45 + {(3 - (2.11))}^{2} \cdot 0.12 + {(4 - (2.11))}^{2} \cdot 0.14}$

Step 7.3

Raise

$- 1.11$ to the power of

$2$ .

$\sqrt{1.2321 \cdot 0.29 + {(2 - (2.11))}^{2} \cdot 0.45 + {(3 - (2.11))}^{2} \cdot 0.12 + {(4 - (2.11))}^{2} \cdot 0.14}$

Step 7.4

Multiply

$1.2321$ by

$0.29$ .

$\sqrt{0.357309 + {(2 - (2.11))}^{2} \cdot 0.45 + {(3 - (2.11))}^{2} \cdot 0.12 + {(4 - (2.11))}^{2} \cdot 0.14}$

Step 7.5

Multiply

$- 1$ by

$2.11$ .

$\sqrt{0.357309 + {(2 - 2.11)}^{2} \cdot 0.45 + {(3 - (2.11))}^{2} \cdot 0.12 + {(4 - (2.11))}^{2} \cdot 0.14}$

Step 7.6

Subtract

$2.11$ from

$2$ .

$\sqrt{0.357309 + {(- 0.11)}^{2} \cdot 0.45 + {(3 - (2.11))}^{2} \cdot 0.12 + {(4 - (2.11))}^{2} \cdot 0.14}$

Step 7.7

Raise

$- 0.11$ to the power of

$2$ .

$\sqrt{0.357309 + 0.0121 \cdot 0.45 + {(3 - (2.11))}^{2} \cdot 0.12 + {(4 - (2.11))}^{2} \cdot 0.14}$

Step 7.8

Multiply

$0.0121$ by

$0.45$ .

$\sqrt{0.357309 + 0.005445 + {(3 - (2.11))}^{2} \cdot 0.12 + {(4 - (2.11))}^{2} \cdot 0.14}$

Step 7.9

Multiply

$- 1$ by

$2.11$ .

$\sqrt{0.357309 + 0.005445 + {(3 - 2.11)}^{2} \cdot 0.12 + {(4 - (2.11))}^{2} \cdot 0.14}$

Step 7.10

Subtract

$2.11$ from

$3$ .

$\sqrt{0.357309 + 0.005445 + {0.89}^{2} \cdot 0.12 + {(4 - (2.11))}^{2} \cdot 0.14}$

Step 7.11

Raise

$0.89$ to the power of

$2$ .

$\sqrt{0.357309 + 0.005445 + 0.7921 \cdot 0.12 + {(4 - (2.11))}^{2} \cdot 0.14}$

Step 7.12

Multiply

$0.7921$ by

$0.12$ .

$\sqrt{0.357309 + 0.005445 + 0.095052 + {(4 - (2.11))}^{2} \cdot 0.14}$

Step 7.13

Multiply

$- 1$ by

$2.11$ .

$\sqrt{0.357309 + 0.005445 + 0.095052 + {(4 - 2.11)}^{2} \cdot 0.14}$

Step 7.14

Subtract

$2.11$ from

$4$ .

$\sqrt{0.357309 + 0.005445 + 0.095052 + {1.89}^{2} \cdot 0.14}$

Step 7.15

Raise

$1.89$ to the power of

$2$ .

$\sqrt{0.357309 + 0.005445 + 0.095052 + 3.5721 \cdot 0.14}$

Step 7.16

Multiply

$3.5721$ by

$0.14$ .

$\sqrt{0.357309 + 0.005445 + 0.095052 + 0.500094}$

Step 7.17

Add

$0.357309$ and

$0.005445$ .

$\sqrt{0.362754 + 0.095052 + 0.500094}$

Step 7.18

Add

$0.362754$ and

$0.095052$ .

$\sqrt{0.457806 + 0.500094}$

Step 7.19

Add

$0.457806$ and

$0.500094$ .

$\sqrt{0.9579}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\sqrt{0.9579}$

Decimal Form:

$0.97872365 \dots$