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Roughly speaking, the integral notation can be thought of as a shorthand for and the differential notation can be thought of as a shorthand for.
If you are not familiar with the Riemann-Stieltjes integral, make sure you also read the lecture entitled Computing the Riemann-Stieltjes integral: some rules , before reading the next example.
Example Let be a random variable with support and distribution function Its expected value is. A completely general and rigorous definition of expected value is based on the Lebesgue integral.
We report it below without further comments. Less technically inclined readers can safely skip it, while interested readers can read more about it in the lecture entitled Expected value and the Lebesgue integral.
Definition Let be a sample space , a probability measure defined on the events of and a random variable defined on. The expected value of is provided the Lebesgue integral of with respect to exists and is well-defined.
An important property of the expected value, known as transformation theorem, allows to easily compute the expected value of a function of a random variable.
Let be a random variable. Let be a real function. Define a new random variable as follows:. Then, provided the above integral exists.
This is an important property. It says that, if you need to compute the expected value of , you do not need to know the support of and its distribution function : you can compute it just by replacing with in the formula for the expected value of.
For discrete random variables the formula becomes while for continuous random variables it is It is possible albeit non-trivial to prove that the above two formulae hold also when is a -dimensional random vector, is a real function of variables and.
When is a discrete random vector and is its joint probability function, then. When is an continuous random vector and is its joint density function, then.
If is a random variable and is another random variable such that where and are two constants, then the following holds:.
For discrete random variables this is proved as follows: For continuous random variables the proof is In general, the linearity property is a consequence of the transformation theorem and of the fact that the Riemann-Stieltjes integral is a linear operator:.
A stronger linearity property holds, which involves two or more random variables. The property can be proved only using the Lebesgue integral see the lecture entitled Expected value and the Lebesgue integral.
The property is as follows: let and be two random variables and let and be two constants; then. Let be a -dimensional random vector and denote its components by , The expected value of , denoted by , is just the vector of the expected values of the components of.
Suppose, for example, that is a row vector; then. Let be a random matrix, i. Denote its -th entry by. The expected value of , denoted by , is just the matrix of the expected values of the entries of :.
Denote the absolute value of a random variable by. If exists and is finite, we say that is an integrable random variable , or just that is integrable.
The space of all random variables such that exists and is finite is denoted by or , where the triple makes the dependence on the underlying probability space explicit.
If belongs to , we write. Hence, if is integrable, we write. Conditional expectation. Introduces the conditional version of the expected value operator.
Properties of the expected value. The expected value is commonly used to indicate the anticipated value of an investment in the future.
On the basis of the probabilities of possible scenarios, the analyst can figure out the expected value of the probable values.
Although the concept of expected value is often used in the case of various multivariate models and scenario analysis, it is predominantly used in the calculation of expected return.
This has been a guide to the Expected Value Formula. Here we learn how to calculate the expected value along with examples and downloadable excel template.
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In the same way as before we can calculate the expected value of games of chance such as roulette. In the U. Half of the are red, half are black.
Both 0 and 00 are green. A ball randomly lands in one of the slots, and bets are placed on where the ball will land.
One of the simplest bets is to wager on red. If the ball lands on a black or green space in the wheel, then you win nothing. What is the expected value on a bet such as this?
Here the house has a slight edge as with all casino games. As another example, consider a lottery. This gives us an expected value of:. So if you were to play the lottery over and over, in the long run, you lose about 92 cents — almost all of your ticket price — each time you play.
All of the above examples look at a discrete random variable. However, it is possible to define the expected value for a continuous random variable as well.