by Marco Taboga, PhD
This lecture discusses some fundamental properties of the expected value operator.
Some of these properties can be proved using the material presented in previous lectures. Others are gathered here for convenience, but can be fully understood only after reading the material presented in subsequent lectures.
It may be a good idea to memorize these properties as they provide essential rules for performing computations that involve the expected value.
Table of contents
Scalar multiplication of a random variable
Sums of random variables
Linear combinations of random variables
Expected value of a constant
Expectation of a product of random variables
Non-linear transformations
Addition of a constant matrix and a matrix with random entries
Multiplication of a constant matrix and a matrix with random entries
Expectation of a positive random variable
Preservation of almost sure inequalities
Solved exercises
Exercise 1
Exercise 2
Exercise 3
Scalar multiplication of a random variable
If is a random variable and is a constant, then
Proof
This property has been discussed in the lecture on the Expected value. It can be proved in several different ways, for example, by using the transformation theorem or the linearity of the Riemann-Stieltjes integral.
Example Let be a random variable with expectationand defineThen,
Sums of random variables
If , , ..., are random variables, then
Proof
See the lecture on the Expected value. The same comments made for the previous property apply.
Example Let and be two random variables with expected valuesand defineThen,
Linear combinations of random variables
If , , ..., are random variables and are constants, then
Proof
This can be trivially obtained by combining the two properties above (scalar multiplication and sum).
Consider as the entries of a vector and , , ..., as the entries of a random vector .
Then, we can also writewhich is a multivariate generalization of the Scalar multiplication property above.
Example Let and be two random variables with expected valuesand defineThen,
Expected value of a constant
A perhaps obvious property is that the expected value of a constant is equal to the constant itself:for any constant .
Proof
This rule is again a consequence of the fact that the expected value is a Riemann-Stieltjes integral and the latter is linear.
Expectation of a product of random variables
Let and be two random variables. In general, there is no easy rule or formula for computing the expected value of their product.
However, if and are statistically independent, then
Proof
See the lecture on statistical independence.
Non-linear transformations
Let be a non-linear function. In general,
However, Jensen's inequality tells us thatif is convex and if is concave.
Example Since is a convex function, we have
Addition of a constant matrix anda matrix with random entries
Let be a random matrix, that is, a matrix whose entries are random variables.
If is a matrix of constants, then
Proof
This is easily proved by applying the linearity properties above to each entry of the random matrix .
Note that a random vector is just a particular instance of a random matrix. So, if is a random vector and is a vector of constants, then
Example Let be a random vector such that its two entries and have expected valuesLet be the following constant vector:DefineThen,
Multiplication of a constantmatrix and a matrix with random entries
Let be a random matrix.
If is a matrix of constants, then
If is a a matrix of constants, then
Proof
These are immediate consequences of the linearity properties above.
By iteratively applying these properties, if is a matrix of constants and is a a matrix of constants, we obtain
Example Let be a random vector such thatwhere and are the two components of . Let be the following matrix of constants:DefineThen,
Expectation of a positive random variable
Let be an integrable random variable defined on a sample space .
Let for all (i.e., is a positive random variable).
Then,
Proof
Intuitively, this is obvious. The expected value of is a weighted average of the values that can take on. But can take on only positive values. Therefore, also its expectation must be positive. Formally, the expected value is the Lebesgue integral of , and can be approximated to any degree of accuracy by positive simple random variables whose Lebesgue integral is positive. Therefore, also the Lebesgue integral of must be positive.
Preservation of almost sure inequalities
Let and be two integrable random variables defined on a sample space .
Let and be such that almost surely. In other words, there exists a zero-probability event such that 
Then,
Proof
Let be a zero-probability event such that First, note thatwhere is the indicator of the event and is the indicator of the complement of . As a consequence, we can write 
By the properties of indicators of zero-probability events, we have Thus, we can writeNow, when , then and . On the contrary, when , then and . Therefore, for all (i.e., is a positive random variable). Thus, by the previous property (expectation of a positive random variable), we have which implies By the linearity of the expected value, we getTherefore,
Solved exercises
Below you can find some exercises with explained solutions.
Exercise 1
Let and be two random variables, having expected values:
Compute the expected value of the random variable defined as follows:
Solution
Using the linearity of the expected value operator, we obtain
Exercise 2
Let be a random vector such that its two entries and have expected values
Let be the following matrix of constants:
Compute the expected value of the random vector defined as follows:
Solution
The linearity property of the expected value applies to the multiplication of a constant matrix and a random vector:
Exercise 3
Let be a matrix with random entries, such that all its entries have expected value equal to .
Let be the following constant vector:
Compute the expected value of the random vector defined as follows:
Solution
The linearity property of the expected value operator applies to the multiplication of a constant vector and a matrix with random entries:
How to cite
Please cite as:
Taboga, Marco (2021). "Properties of the expected value", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/fundamentals-of-probability/expected-value-properties.
