Finite Math Examples

\begin{matrix} x & P (x) - 1 & \frac{3}{2} 0 & \frac{1}{2} 1 & \frac{1}{6} 2 & \frac{1}{18} 3 & \frac{1}{54} \end{matrix}

$\begin{array}{c} x & P (x) \\ - 1 & \frac{3}{2} \\ 0 & \frac{1}{2} \\ 1 & \frac{1}{6} \\ 2 & \frac{1}{18} \\ 3 & \frac{1}{54} \end{array}$

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

\frac{3}{2}

$\frac{3}{2}$ is not less than or equal to

1

$1$ , which doesn't meet the first property of the probability distribution.

\frac{3}{2}

$\frac{3}{2}$ is not less than or equal to

1

$1$

Step 1.3

\frac{1}{2}

$\frac{1}{2}$ is between

0

$0$ and

1

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

\frac{1}{2}

$\frac{1}{2}$ is between

0

$0$ and

1

$1$ inclusive

Step 1.4

\frac{1}{6}

$\frac{1}{6}$ is between

0

$0$ and

1

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

\frac{1}{6}

$\frac{1}{6}$ is between

0

$0$ and

1

$1$ inclusive

Step 1.5

\frac{1}{18}

$\frac{1}{18}$ is between

0

$0$ and

1

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

\frac{1}{18}

$\frac{1}{18}$ is between

0

$0$ and

1

$1$ inclusive

Step 1.6

\frac{1}{54}

$\frac{1}{54}$ is between

0

$0$ and

1

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

\frac{1}{54}

$\frac{1}{54}$ is between

0

$0$ and

1

$1$ inclusive

Step 1.7

The probability

P (x)

$P (x)$ does not fall between

0

$0$ and

1

$1$ inclusive for all

x

$x$ values, which does not meet the first property of the probability distribution.

The table does not satisfy the two properties of a probability distribution

Step 2

The table does not satisfy the two properties of a probability distribution, which means that the standard deviation can't be found using the given table.

Can't find the standard deviation