# Statistics Examples

Step 1
Prove that the given table satisfies the two properties needed for a probability distribution.
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
Step 1.8.2
Step 1.8.3
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
Simplify each term.
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
Step 4.2
Step 4.3
Step 5
The standard deviation of a distribution is a measure of the dispersion and is equal to the square root of the variance.
Step 6
Fill in the known values.
Step 7
Simplify the expression.
Step 7.1
Multiply by .
Step 7.2
Subtract from .
Step 7.3
Raise to the power of .
Step 7.4
Multiply by .
Step 7.5
Multiply by .
Step 7.6
Subtract from .
Step 7.7
Raising to any positive power yields .
Step 7.8
Multiply by .
Step 7.9
Multiply by .
Step 7.10
Subtract from .
Step 7.11
One to any power is one.
Step 7.12
Multiply by .
Step 7.13
Multiply by .
Step 7.14
Subtract from .
Step 7.15
Raise to the power of .
Step 7.16
Multiply by .
Step 7.17