Contents
What is the expectation of an integral?
Generally speaking, the expected value of an integral is an iterated integral, and so the normal mathematical rules for interchange of integrals apply. To see this more clearly, we first note that the expectation operator is an integration operation.
When can we interchange summation and integration?
Exercise: verify that integration with respect to counting measure is the same as summation, where the integral exists and is finite iff the sum converges absolutely. (That is, given a sequence of real numbers an, define a function b:N→R by b(n)=an. Then ∫Nbdν=∑∞n=1an.)
Can we interchange summation and integration?
the dominated convergence theorem implies that we may integrate the sequence of partial sums term-by-term, which is tantamount to saying that we may switch integration and summation. The idea behind the method is to pick our g ‘s as simple as possible so that it is easy to integrate them and apply the criterion.
What is the difference between summation and integration?
What is the difference between summation and integration? Summation involves the discrete values with the upper and lower bounds, whereas the integration involves continuous value. Summation is a process of discrete addition while Integration is a process of adding continuous things.
Is it true if expectation and integral term can be interchanged?
$\\begingroup$@ Jonas : no it is not always true, but if you can interchange expectation and integral term then it is true so you only have to derive the conditions under which such operation is ok. Regards.$\\endgroup$– TheBridgeOct 22 ’12 at 20:58 3
How is interchanging a derivative with an expectation done?
Interchanging a derivative with an expectation or an integral can be done using the dominated convergence theorem. Here is a version of such a result. Lemma.
When can we interchange the integral and summation?
When can we interchange the integral and summation? Is f n ( x) ≥ 0 for all x and for all n sufficient? How about when ∑ f n ( x) converges absolutely? If so why? Theorem. If { f n } n is a positive sequence of integrable functions and f = ∑ n f n then ∫ f = ∑ n ∫ f n. Proof. Consider first two functions, f 1 and f 2.
When do you interchange integration and differentiation Leibniz?
You may interchange integration and differentiation precisely when Leibniz says you may. In your notation, for Riemann integrals: when f and ∂f ( x, t) ∂x are continuous in x and t (both) in an open neighborhood of {x} × [a, b]. There is a similar statement for Lebesgue integrals.