Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
Intro Physics Homework Help
Advanced Physics Homework Help
Precalculus Homework Help
Calculus Homework Help
Bio/Chem Homework Help
Engineering Homework Help
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
Intro Physics Homework Help
Advanced Physics Homework Help
Precalculus Homework Help
Calculus Homework Help
Bio/Chem Homework Help
Engineering Homework Help
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Homework Help
Engineering and Comp Sci Homework Help
Probability: Proof of E[X/X+Y]=(X+Y)/2, X and Y are i.i.d. r.v.
Reply to thread
Message
[QUOTE="fzero, post: 4527953, member: 274413"] This seems weird to me, both for the reason you mention, that ##E[f(X,Y)]## can't be a function of the random variables, and also because you can easily show that $$ E[X/(X+Y)] + E[Y/(X+Y)] = 1$$ for a normalized distribution. So the only way we can have ##E[X/(X+Y)] = E[Y/(X+Y)]## is if they are both 1/2. Aside from those concerns, one way to approach the reciprocal of a random variable is through a power series. Namely, given a random variable ##X##, we can define a new random variable ##X'## by $$ X = \bar{X} + X',~~~~ E[X']=0,$$ where ## \bar{X} = E[X]##. Then, if ##\bar{X}\neq 0##, we can use the geometric series to define $$ \frac{1}{X} = \frac{1}{\bar{X}} \sum_{n\geq 0} (-1)^n \left( \frac{X'}{\bar{X}} \right)^n ,$$ which is convergent if ##|X'| \leq |\bar{X}|##. A similar construction would serve to define ##1/(X+Y)## or similar rational functions. If you weren't already given a seemingly misleading answer to the question, this is the approach I would suggest. [/QUOTE]
Insert quotes…
Post reply
Forums
Homework Help
Engineering and Comp Sci Homework Help
Probability: Proof of E[X/X+Y]=(X+Y)/2, X and Y are i.i.d. r.v.
Back
Top