Is the expected value of an integral integrable?

Is the expected value of an integral integrable?

Thus, as with integrals generally, an expected value can exist as a number in R (in which case X is integrable ), can exist as ∞ or − ∞ , or can fail to exist. In reference to part (a), a random variable with a finite set of values in R is a simple function in the terminology of general integration.

How is the expected value of a variable defined?

In the introductory section, we defined expected value separately for discrete, continuous, and mixed distributions, using density functions. In the section on additional properties, we showed how these definitions can be unified, by first defining expected value for nonnegative random variables in terms of the right-tail distribution function.

Do you need to know the properties of the integral?

However, to understand the exposition, you will need to review the advanced sections on the integral with respect to a positive measure and the properties of the integral. If you are a new student of probability, or are not interested in the measure-theoretic detail of the subject, you can safely skip this section.

Is the expected value of g ( x ) measurable?

If g: S → R is measurable, then g(X) is a real-valued random variable. The following result shows how to computed the expected value of g(X) as an integral with respect to the distribution of X , and is known as the change of variables theorem.

How is the expected value of X defined?

If X is a real-valued random variable on the probability space, the expected value of X is defined as the integral of X with respect to P, assuming that the integral exists: E(X) = ∫ΩXdP Let’s review how the integral is defined in stages, but now using the notation of probability theory.

What is the hyper ellipse of a multivariate normal distribution?

Recall the Multivariate Normal Density function below: You will note that this density function, ϕ ( x), only depends on x through the squared Mahalanobis distance: This is the equation for a hyper-ellipse centered at μ. For a bivariate normal, where p = 2 variables, we have an ellipse as shown in the plot below:

How to calculate the expected value of a random variable?

If X is a random variable and A is an event with P(A) = 0 . Then E(X; A) = 0 . If X is a random variable whose expected value exists, and Y is a random variable with P(X = Y) = 1 , then E(X) = E(Y) .

How to calculate the expected value of g ( x )?

The following result shows how to computed the expected value of g(X) as an integral with respect to the distribution of X , and is known as the change of variables theorem. If g: S → R is measurable then, assuming that the expected value exists, E[g(X)] = ∫Sg(x)dP(x)

How to integrate an absolute value into an integral?

Integrating an Absolute Value Z 4 0 jx3 5×2 + 6xjdx There is no anti-derivative for an absolute value; however, we know it’s de nition. jxj= ˆ x if x 0 x elsewise Thus we can split up our integral depending on where x3 5×2 + 6x is non-negative. x3 5×2 + 6x 0: x(x2 5x+ 6) 0: x(x 2)(x 3) 0:

Which is the expected value of a function?

Recall that this means that μ(A) = 0 implies P(A) = P(X ∈ A) = 0 for A ∈ S . By the Radon-Nikodym theorem, named for Johann Radon and Otto Nikodym, X has a probability density function f with respect to μ .