Homework Statement
Show that the operator on L^2(0,\infty) defined by g \rightarrow f(x)= \int_{0}^{\infty} e^{-xy}g(y)dy is bounded.
Homework Equations
Operator norm: ||T|| = \sup_{||g||_{L^2}=1}||Tg||_{L^2}
The Attempt at a Solution
I tried to get a handle on f(x)=...
Homework Statement
X_1 , \dots ,X_n \sim U[0, \theta] iid. \theta \sim U[0,1]
derive the Bayesian posterior mean estimator
Homework Equations
f(\theta |\vec{X}) = \frac{f( \vec{X}|\theta)f(\theta )}{f( \vec{X})}
The Attempt at a Solution
My line of thinking...
First, the marginal...
I know what I need to show to prove that T=t. i.e., O in T => O in t, since we are given t a subset of T. I just don't see how to do this with the information given.
I guess I should have been more clear with what I need help with. I just am missing whatever key observation I need to make...
actually it is enough to know that if t is a subset of T and X is compact with T, then it is compact with t.
Choose any open covering for (X, t), {Ui} for i in some index I, then it is also and open of (X,T), since t a subset of T. Since (X,T) compact, there is a finite open cover, {Ui}...
Homework Statement
Let X be a set and t & T be two topologies on X. Prove that if (X,t) is Hausdorff and (X, T) is Compact with t a subset of T, then t=T. (i.e., T is a subset of t).The Attempt at a Solution
potentially useful theorem: (X,t) Hausdorff and X compact implies that each subset F...
A water distribution company in southern California gets its water supply from the north and sell it back to its customers in Orange county. Assume the following simplified scheme: 3 MG (millions of gallons) of water arrives from the north at the beginning of the month. The company can store up...
A total of r keys are to be put one at a time, in k boxes, with each key
independently being put in box i with probability pi ,
∑pi = 1. Each time a key is put in a nonempty box, we say that a collision occurs. Find
the expected number of collisions.
My professor hinted that we should use...
yeah, it was really late when i pieced those ideas together, so I didn't think they were exactly right...I'll post the "solution" when they are posted if you're interested...
If you really think that is the case I suggest you look at modern economics again...in particular I might suggest Microeconomic Theory by Mas-Colell...or maybe
Recursive Methods in Economic Dynamics by Stokey, Lucas, Prescott
both standard PhD Micro/Macro books...
I think this is what I'm going to say...
suppose to the contrary that e^x is not convex,
i.e., suppose there exists a t in the closed interval 0 to 1 such that
t * exp[x1] + (1-t) * exp[x2] < exp[tx1 +(1-t)x2]. also assume wlog that x2<x1
note that when t=0 and t=1, the two equations are...
hmmm, well, I'm an undergrad in a PhD microeconomics class...though only about half of the grad students have had analysis (I'm in the intro to analysis class now...)
what you said makes sense...and I thought something along those lines, though didn't really know how to phrase it into a...
I know that if I can get 1 I can get 2, and vice-versa. The problem is I can't seem to get either...at least in terms of a mathematical proof. I can look at the graph and see that the set is convex...but that won't suffice on my HW.
Homework Statement
prove the set is convex or give a counter example:
1. S = {(x,y) : y >/= e^x}
2. T = {(x,y) : y </= ln(x)}
Homework Equations
defn. of convexity
(x1, y1) and (x2, y2) in S => (tx1 +(1-t)x2, ty1 + (1-t)y2) in S, 0 </= t </= 1
The Attempt at a Solution
I think 2 will be...