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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

    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...
  4. 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...
  5. 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...
  6. 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...
  7. 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...
  8. 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...
  9. 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(...
  10. 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...
  11. 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...
  12. 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...
  13. 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
  14. 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...
  15. 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...
  16. M

    Show that the inequality is true | Geometric Mean

    Homework Statement Let r_{1}, r_{2}, ... , r_{n} be strictly positive numbers. Show that the inequality (1+R_{G})^{n} \leq V is true. Where R_{G} = (r_{1}r_{2}...r_{n})^{1/n} and V= \Pi_{k=1}^{n} (1+r_{k}) Homework Equations The Attempt at a Solution I've...
  17. M

    Prove this inequality : Geometric Mean and Arithmetic Mean

    Homework Statement let r_{1}, r_{2}, ... , r_{n} be strictly positive numbers. Suppose an investment of one dollar at the beginning of the year k grows to 1+r_{k} at the end of year k (so that r_{k} is the "return on investment" in year k). Then the value of an investment of one dollar at...
  18. M

    Two variable limit problem : Polar Coordinates

    Homework Statement Find the limit of lim_{(x,y) \rightarrow (0,0)} xy(\frac{x^{2}-y^{2}}{x^{2}+y^{2}}) Homework Equations The Attempt at a Solution We were supposed to switch to polar coordinates to solve this problem. Thus we get, lim_{(r) \rightarrow (0)} rcos\theta rsin\theta...
  19. M

    Show that u'(t) = r(t).(r'(t)Xr'''(t))

    Homework Statement Let u(t) be a vector valued function, where u(t) = r(t).(r'(t)Xr''(t)) where r(t) is a vector valued function, and (r'(t)Xr''(t)) the cross product of the first and second derivative of r(t). Show that u'(t) = r(t).(r'(t)Xr'''(t)) where r'''(t) is the 3rd...
  20. M

    Why is the gamma function equal to (n-1)! ?

    Homework Statement Why is the equality below true? \Gamma(n) = (n-1)! Where \Gamma(n) = \int^{\infty}_{0} x^{n-1} e^{-x}dx Homework Equations The Attempt at a Solution I've read the article on wikipedia but I cannot understand it. Is there any special properties in calculus that I must...
  21. M

    Factorial : n!/(n-k)! = n(n-1)(n-2) (n-k+1) - why?

    Why is the equation (A) n!/(n-k)! = n(n-1)(n-2)...(n-k+1) true? For example, let n=4 and k=2, then 4!/2! = 4x3x2x1 / 2x1 = 4x3 = 12. I understand this example, but I can't make the connection with this and the right-hand-side of equation (A). For example, why is our...
  22. M

    Definite Integral: Exponential

    Homework Statement \int^{\infty}_{0} e^{-y} dy = 1 Homework Equations The Attempt at a Solution Why is this equality true? I understand that the integral and derivative of e^y is always e^y, but I can't make out why this definite integral is equal to 1. I graphed it using a...
  23. M

    Countable sets | If k:A->N is 1-to-1, then A is

    Countable sets | If k:A-->N is 1-to-1, then A is... Homework Statement Suppose we found a 1-to-1 function k that maps the set A to the set N, where N is the set of natural numbers. What can we say about the set A? Homework Equations The Attempt at a Solution The answer is 'A is...
  24. M

    Whats the difference between a definition and an axiom?

    What is it? For example A function f: A-->B is called injective if, for all a and a' in A, f(a)=f(a') implies that a=a'. What is keeping this definition from being an axiom?
  25. M

    Set of all the limit points of Set E. Prove that its closed.

    Is it correct to make the following statement? If a point x in E is not a limit point of E, then any neighborhood V of x will--at most--contain finitely many points of E. Thus, its possible for V to contain only one point, namely, x. Thanks, M
  26. M

    An example of a close and bounded set that is not compact

    Take the discreet metric on an infinite set A. I understand that its closed (because it contains all of its limit points), but I don't understand why its bounded and why its not compact. Also, when they say "an infinite set A" do they mean a set that extends to infinite (say, [1,n] for...
  27. M

    Infinite intersection of open sets

    I understand that the finite intersection of open set is open, but is it true that the infinite intersection of open set is closed? or is it possible for it to be open as well? Thank you, M
  28. M

    Logical Implication - If p, then q

    If p, then q. Suppose p is false but q is true. Why is it that the implication "If p, then q" is still true? For example, If x=2, then x + 3 = 5. Suppose x is NOT 2 (i.e. p is false), but x+3=5 (q is still true). Why is the implication "If x=2, then x + 3 = 5" still true? Is the...
  29. M

    Integration by parts

    Integration by parts - Exponential distribution Homework Statement Solve the following definite integral: \int^{\infty}_{0} \frac{1}{\lambda} x e^{-\frac{x}{\lambda}} dx I'm asked to solve this integral. The solution is \lambda, although I'm not sure how this was done. Homework...
  30. M

    Statistics: SSE, MSE, R^2, and C-statistic | Fill in the blank puzzle

    Hello, I'm studying for a final exam and I'm having trouble with this particular question. I attached the data required to solve the problem. I am required to fill in the missing information in the data. I've defined the acronyms below: SSE = Sum of Square Errors MSE = Mean Square of...
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