Eroxl's Notes
Mean

The mean of a distribution is it's long-running average value. Formally the mean is defined as the expected value which is calculated using the following formulae.

Formula

Discrete Case

For a discrete random variable (ie. a randomized dice that can only be the integers 1-6):

Where the sum is taken over all possible values of and is the probability that the random variable takes the value of .

Continuous Case

For a continuous random variable:

Estimations

The mean of a distribution can be estimated be estimated when the exact probabilities of values are unknown.

Sample Survey

Given a sample survey with being the number of observation and being the observation we can estimate the mean of the random variable as follows:

Example

The numbers of hours spent studying for a subset of students are 4, 6, 8, 7, 5. Estimate the mean number of hours spent studying for students.

Properties

Addition by a Constant

Multiplication by a Constant

Addition

Multiplication