I have a question about the notation used in this problem please dont solve the problem for me. Sho

cyfwelestoi 2022-05-29 Answered
I have a question about the notation used in this problem please dont solve the problem for me.
Show that | | X Y | | 1 = | | X | | 1 | | Y | | 1 for independent r.vs X and Y. Show further that if X and Y are also integrable, then E ( X Y ) = E ( X ) E ( Y )
I don't understand what the difference between | | X | | 1 and E ( | X | ). I assume we're talking about the L 1 ( P ) norm? But then
| | X | | 1 = Ω | X ( ω ) | d P ( ω ) = E ( | X | )
So if this is finite, then E ( X ) exists therefore it is integrable already (so why "also"?) Therefore the first "show that" would follow from the second
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Elbrinidr
Answered 2022-05-30 Author has 9 answers
Under independence | | X Y | | 1 = | | X | | 1 | | Y | | 1 holds even if E | X | or E | Y | is infinity (by Tonelli's Theorem). But E ( X Y ) = ( E X ) ( E Y ) holds if the expectations are finite.
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-02-05
What is the algebraic expression for: Five less than the quotient of x and y equals 25?
asked 2022-06-08
a) Identify equations, expressions and inequalities
b) Consider
5 = 3 m
n > | 8 |
2 + 6 = ( 5 1 ) × 2
8 × ( k + 6 )
asked 2021-10-26
Simplify the expression
(2x3)5(2x4)4=2
asked 2022-08-03
Solve
1. | x | = 7
2. | x | + 4 = 6
3. | m 5 | = 4
4. | m + 9 | = 3
asked 2022-06-13
Let X be an arbitrary space, and let μ, ν be two probability measures on X. Suppose there is a common set A on which μ ( A ) = ν ( A ) = 1.
Now suppose there are B , C such that μ ( B ) = ν ( C ) = 1. Can anything be said of μ ( C ) and ν ( B ), or μ ( B C ) and ν ( B C )?
My initial intuition was that the answer should be yes since B A and C A both have full μ and full ν measure respectively, and so there should be "enough" overlap for B C to have at least positive measure. But I can't seem to prove this - so I'm interested in whether anything can be said of these quantities (either alone or possibly together, e.g. is it possible for μ ( C ) = ν ( B ) = 0) and whether there is any clever counterexample for which μ ( C ) and ν ( B ) can be any arbitrary number in [0,1].
asked 2022-06-16
How do you solve x - 14 23 ?
asked 2021-12-17
Solve, please, x=b-cd for c.

New questions