Statistics 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

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 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 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 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 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 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 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 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 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 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 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 9.3

Add

0.7

$0.7$ and

0.1

$0.1$ .

0.8 + 0.2

$0.8 + 0.2$

Step 9.4

Add

0.8

$0.8$ and

0.2

$0.2$ .

1

$1$

1

$1$

Step 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

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

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