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  1. M

    A simple conditional expectation question

    Let v be a random variable distributed according to F(.). Let X be a set containing the objects x1 and x2. Suppose E(v|x1) = b AND E(v|x2) = b (The expected value of v conditional on x1 is b, etc) where b is some constant. Does it follow that E(v|x1,x2) = b? If so, why...
  2. M

    How do you graph this function?

    Homework Statement For a given fixed value, z, how would the level curves of the function below look like? (i.e. the graph of the function in the xy plane, for some given value z. z = min(2x+1, x+y, 2y+1) Homework Equations The Attempt at a Solution Ive been working on...
  3. M

    Given a discontinuos function, show that it is not concave

    Its called Microeconomic Theory by Mas-Collel, Winston, and Green. I thought concavity was a general mathematical property, I learned it in Real Analysis.
  4. M

    Given a discontinuos function, show that it is not concave

    I'm trying to show that if the function f is discontinuous, then it cannot be concave. Here's the general definition of concavity Let f be a function of many variables defined on the convex set S. Then f is concave on the set S if for all x ∈ S, all x' ∈ S, and all λ ∈ (0,1) we have...
  5. M

    Given a discontinuos function, show that it is not concave

    Hello, It's a class in Microeconomic Theory Thanks
  6. M

    Given a discontinuos function, show that it is not concave

    Homework Statement Let f be a function from (1,0) to (1,0). Suppose that f is discontinuous. Show that f is not concave. Homework Equations The Attempt at a Solution Let f:(0,1)-->(0,1). Suppose f is discontinous. Show that it is not concave. I've been working on this problem for over...
  7. M

    Traveling at Light Speed Through Space: A Thought Experiment

    Thanks everyone for your response. I'm an amateur in physics so could someone explain why it is not theoretically possible for a spaceship to travel at the speed of light? Does it have something to do with Einstein's E=mc^2? Also, does ".99c" stand for ".99 percent of the speed of light"...
  8. M

    Traveling at Light Speed Through Space: A Thought Experiment

    My brother-in-law proposed the following thought experiment: Suppose a person was sitting on the nose of a spaceship traveling at the speed of light through outer space. Now suppose that person pushed off against the spaceship launching himself ahead of it. Is it the case that that person will...
  9. M

    Modulo operation | What does this mean?

    Homework Statement Suppose i and j take on values from {0,1,2,...,7}. We say that i and j are 'happy' if i - j (their difference) is congruent to 1.4 or 7 modulo 8. Note: 'happy' is some mathematical property not relevant to the question. What does 7 modulo 8 mean? What does it...
  10. M

    Size of the Power Set

    Homework Statement Why is the size of the power set 2^n ? To elaborate, suppose we have a set B that has n elements. Let C be the set that contains all the possible subset of B. Why is it that |C|=2^n ? It boggles my mind why the base is 2 for all size of sets. Thank you, M...
  11. M

    Whats the derivative of the absolute value to the power of p?

    Ok, |x|^p is definitely continuous everywhere. I graphed it and it looks differentiable at x=0 (seems to be the local min). Is the following a valid proof? Show that |x|^p, p>1 is differentiable at x=0 lim_{h->0^{+}} \frac{|0+h|^{p} - |0|^p}{h}= lim_{h->0^{+}} \frac{|h|^{p}}{h}=lim_{h->0^{+}}...
  12. M

    Whats the derivative of the absolute value to the power of p?

    Homework Statement Let p>0. What is the derivative of |x|^{p}? Homework Equations The Attempt at a Solution I know that if p is even, then the derivative is just px^{p-1} . But what if x is odd? Would it turn out to be some piecewise function, such as px^{p-1},\: if x \geq 0 and...
  13. M

    Prove that the functin is differentiable at (0, ,0).

    You mean using: lim_{\textbf{h} \rightarrow \textbf{0}} \frac{f(\textbf{0} + \textbf{h}) - f(\textbf{0}) - Df(\textbf{a})(\textbf{h})}{||\textbf{h}||} = \textbf{0} . directly?
  14. M

    Prove that the functin is differentiable at (0, ,0).

    Hi Euler, Thank you for your reply. There's a theorem in my textbook that says that if the first-order partial derivatives of a vector-valued function f exists at a, and if these first-order partial derivatives are continuous at a, then f is differentiable at a. I know how to show the...
  15. M

    Prove that the functin is differentiable at (0, ,0).

    Homework Statement Let r>0, and let f be a function from B_{r}(\textbf{0}) \rightarrow \textbf{R} , and suppose that there exists an \alpha > 1 such that |f(\textbf{x})| \leq ||\textbf{x}||^{\alpha} for all \textbf{x} \in B_{r}(\textbf{0}). Prove that f is differentiable at 0. What happens...
  16. M

    A simple thought experiment | Please confirm

    Ah of course...about your second point, my instinct tells me that the other two will feel the pressure created by the enlargement of the stomach of the first person. But they won't move because the force created by the enlargement of the stomach will not over power the force exerted by the leg...
  17. M

    A simple thought experiment | Please confirm

    Please excuse my naivete, but this thought experiment has been running through my mind all week. Suppose you have three people in room. Suppose the room is sealed such that no air can come in and no air can come out. Suppose there's enough oxygen in the room for us to conduct our experiment(...
  18. M

    Maximum and Minimum : Langrange multiplier problem

    Homework Statement Find the maximum and minimum of the function f over the closed and bounded set S. Use langrange multiplier method to find the values of the boundary points. Homework Equations f(x,y) = (1+x+y)2 S = {(x,y) : x2/4 + y2/16 <= 1} The Attempt at a Solution...
  19. M

    Is this equality generally true? | Probability

    Suppose Xn is a random variable. Let b and c be a constant. Is the following generally true? P(|X_{n}-b| \geq \epsilon) = P(|X_{n}-b|^{2} \geq \epsilon^{2}) This says that the probability that Xn minus b is greater than or equal to epsilon is equal to the probability that Xn minus b...
  20. M

    Simple word problem: Chain rule

    Thanks, would you be so kind as to elaborate on your answer? I would appreciate it. My background in trigonometry is weak; although I do know how to use them on a right triangle, but in this case, I'm not quite certain.
  21. M

    Simple word problem: Chain rule

    Homework Statement One side of a triangle is increasing at a rate of 3cm/s and a second side is decreasing at a rate of 2cm/s. If the area of the triangle remain constant, at what rate does the angle between the sides change when the first side is 20cm long and the second side is 30cm, and...
  22. M

    Probability inequality : Is the following always true?

    Homework Statement P(AUB) <= P(A) + P(B) Homework Equations The Attempt at a Solution I can't understand the intuition behind this property. It's not a homework assignment, it was just something that came up in class. Thanks, M
  23. M

    Prove the equality : Multivariable chain rule problem

    Homework Statement Prove that (\frac{\partial u}{\partial x})^{2} + (\frac{\partial u}{\partial t})^{2} = e^{-2s}[(\frac{\partial u}{\partial s})^{2} + (\frac{\partial u}{\partial t})^{2}]. Homework Equations u = f(x,y) x = e^{s}cost y = e^{s}sint The Attempt at a Solution I started out...
  24. M

    Prove that F is discontinuous at every rational number

    This means that F(y3-)<F(y3+), since F(y3+) will be greater by 1/2^3. This implies a discontinuity at y3 (where y is a rational). Now I'm trying to prove that F is continuous at each rational. I understand it heuristically, I just need to write it out in symbols.
  25. M

    Prove that F is discontinuous at every rational number

    Hi Coto, Thanks for the reply. I understand the first paragraph clearly. So for a given xk, as x approach from the left, H = 0. Whereas as x approach from the right H=1. Now for the second paragraph, for a given rational q, this rational will also appear in the sequence x1, x2, x3...
  26. M

    Prove that F is discontinuous at every rational number

    I don't think I understand what you mean. So at some rational number y, we have the sum of H(y-xk). Now if we are approaching y from the left, then, depending on where the sequence terms xk is, H will be either 1 or 0. Now as we are approaching from the right, the opposite will happen...
  27. M

    Prove that F is discontinuous at every rational number

    Homework Statement Let x_{1}, x_{2}, ... be a sequence of rational numbers in which each rational number in (0,1) occurs exactly once. Define the function, H(x) = 0 if x \leq 0, and 1 if x > 0. Next, define the function F(x)= \sum^{\infty}_{k=1} 2^{-k} H(x - x_{k}). Prove that F is...
  28. M

    Show that the inequality is true | Geometric Mean

    Any other insights? The prof hinted that we should use log(1+e^x) and associate r with e^x.
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