# Statistics Examples

Step 1

Step 1.1

A discrete random variable takes a set of separate values (such as , , ...). Its probability distribution assigns a probability to each possible value . For each , the probability falls between and inclusive and the sum of the probabilities for all the possible values equals to .

1. For each , .

2. .

Step 1.2

is between and inclusive, which meets the first property of the probability distribution.

is between and inclusive

Step 1.3

is between and inclusive, which meets the first property of the probability distribution.

is between and inclusive

Step 1.4

is between and inclusive, which meets the first property of the probability distribution.

is between and inclusive

Step 1.5

is between and inclusive, which meets the first property of the probability distribution.

is between and inclusive

Step 1.6

For each , the probability falls between and inclusive, which meets the first property of the probability distribution.

for all x values

Step 1.7

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

Step 1.8

The sum of the probabilities for all the possible values is .

Step 1.8.1

Add and .

Step 1.8.2

Add and .

Step 1.8.3

Add and .

Step 1.9

For each , the probability of falls between and inclusive. In addition, the sum of the probabilities for all the possible equals , 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: for all values

Property 2:

The table satisfies the two properties of a probability distribution:

Property 1: for all values

Property 2:

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.

Step 3

Step 3.1

Multiply by .

Step 3.2

Multiply by .

Step 3.3

Multiply by .

Step 3.4

Multiply by .

Step 4

Step 4.1

Add and .

Step 4.2

Add and .

Step 4.3

Add and .

Step 5

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

Step 6

Fill in the known values.

Step 7

Step 7.1

Simplify each term.

Step 7.1.1

Multiply by .

Step 7.1.2

Subtract from .

Step 7.1.3

Raise to the power of .

Step 7.1.4

Multiply by .

Step 7.1.5

Multiply by .

Step 7.1.6

Subtract from .

Step 7.1.7

Raise to the power of .

Step 7.1.8

Multiply by .

Step 7.1.9

Multiply by .

Step 7.1.10

Subtract from .

Step 7.1.11

Raise to the power of .

Step 7.1.12

Multiply by .

Step 7.1.13

Multiply by .

Step 7.1.14

Subtract from .

Step 7.1.15

Raise to the power of .

Step 7.1.16

Multiply by .

Step 7.2

Simplify by adding numbers.

Step 7.2.1

Add and .

Step 7.2.2

Add and .

Step 7.2.3

Add and .