Search results

1. Maximum Entropy Distribution Given Marginals

Hi all, I'm a pure mathematician (Graph Theory) who has to go through a Physics paper, and I am having trouble getting through a part of it. Maybe you guys can point me in the right direction: Let P(x,y) be a joint distribution function. Let H = - \Sigmax,y P(x,y) log P(x,y), which is...
2. Lagrange Multiplier Help

Homework Statement L = - \Sigma x,y (P(x,y) log P(x,y)) + \lambda \Sigmay (P(x,y) - q(x)) This is the Lagrangian. I need to maximize the first term in the sum with respect to P(x,y), subject to the constraint in the second term. The first term is a sum over all possible values of x,y...
3. Unbiased Estimator for Gamma Distribution Parameter

Homework Statement I need to show that c / (sample mean X) is unbiased for b, for some c. Where {X} are iid Gamma (a,b). Homework Equations N.A. The Attempt at a Solution I know how to show that (sample mean X) / a is unbiased for 1/b : 1) E(sample mean X / a) =...
4. Counting Outcomes - Probability Question

Homework Statement Z plays a game where independent flips of a coin are recorded until two heads in succession are encountered. Z wins if 2 heads in succession occurs. Z loses if after 5 flips, we have not encounter two heads in succession. 1) What is the probability that Z wins the game...
5. Probability Mass Function of { Z | Z < 1 }, Z given Z is less than 1

Probability Mass Function of { Z | Z < 1 }, "Z given Z is less than 1" Homework Statement Given Z = X + Y. Find the probability density function of Z|Z < 1. Homework Equations N.A. The Attempt at a Solution f(z) = P(Z=z|Z<1) = P(Z=z AND Z < 1) / P(Z < 1). I thought the...
6. Density Function for Sums of Random Variables

Homework Statement Given the joint density, f(x,y), derive the probability density function for Z = X + Y and V = Y - X. Homework Equations f(x,y) = 2 for 0 < x < y < 1 f(x,y) = 0 otherwise. The Attempt at a Solution For Z = X + Y, I can derive the fact that, f_Z(z) =...
7. Support of Continuous Conditional Density Functions (Probability)

f(x,y) = x + y is the joint probability density function for continuous random variables X and Y. The support of this function is {0 < x < 1, 0 < y < 1}, which means it takes positive values over this region and zero elsewhere. g(x) = x + (1/2) is the probability density function of X...
8. Jacobian of the linear transform Y = AX

Homework Statement Y = AX = g(X) Where X,Y are elements of R^n and A is a nxn matrix. What is the Jacobian of this transformation, Jg(x)? Homework Equations N.A. The Attempt at a Solution Well, I know what to do in the non-matrix case. For example... U = g(x,y) V =...
9. Deriving Joint Distribution Function from Joint Density (check my working pls!)

Homework Statement Given: f(x,y) = x + y, for 0<x<1 and 0<y<1 f(x,y) = 0, otherwise Derive the joint distribution function of X and Y. Homework Equations N.A. The Attempt at a Solution Using the definition, I obtained part of the joint distribution F(x,y) = (1/2)(xy)(x+y)...
10. Sum of two continuous uniform random variables.

Z = X + Y Where X and Y are continuous random variables defined on [0,1] with a continuous uniform distribution. I know we define the density of Z, fz as the convolution of fx and fy but I have no idea why to evaluate the convolution integral, we consider the intervals [0,z] and [1,z-1]...
11. Probability - Infinite Union of Subsets of a Sample Space

Homework Statement This is a question about mathematical probability, using the sigma-algebra, measure and probability space approach. Define A(t) = {all outcomes, w, in the sample space such that Y(w) < or = t} where Y is a random variable and t is any real number. Fix a real number...
12. Statistics Question: The 3rd Moment of Poisson Distribution

Homework Statement X is a discrete random variable that has a Poisson Distribution with parameter L. Hence, the discrete mass function is f(x) = L^{x} e^{-L} / x!. Where L is a real constant, e is the exponential symbol and x! is x factorial. Without using generating functions, what is...
13. Notation - f(x,a) vs f(x;a)

Homework Statement Quick question... I have seen both being used : f(x,a) and f(x;a). What is the usual convention? Are both acceptable to denote functions of 2 variables (in this case f is a function of both x and a). Or are there vital differences between the two that I don't know...
14. Time Travel Question (from A Briefer History of Time )

Time Travel Question (from "A Briefer History of Time") I read the original "A Brief History of Time" and is now reading "A Briefer History of Time". :P There is a part about time travel that confuses me. Does anyone want to enlighten me using layman terms? (i am a mathematics major, not a...
15. Mathematical Probability Question

I spent quite a lot of time trying to figure out this question that I read in a book. Through trial and error, I got a rough idea of what the solution should be like but can't figure out how I would derive it mathematically. Can anyone give me some directions? :) QUESTION: n people stand...
16. Limits of Functions at Infinity

I read that "if f : R -> R is an increasing function, then limit as x tend to infinity of f(x) is either infinity, minus infinity or a real number". f an increasing function means { x < y } => { f(x) < or = f(y) }. How do I prove this (if it is true)? Can I apply this to a function g : R ->...
17. Another Polar Coordinates + Integration Question

I came across this example on the net : We are integrating over the region that is the area inside of r = 3 + 2 sin θ and outside of r = 2, working in polar coordinates (r,θ). What is the limits of integration for θ? # I already know the answer. But I have no idea how to arrive at the...
18. Need help with Polar Coordinates, esp. for integration

Any web resources regarding changing the variable of integration from cartesian to polar coordinates that goes beyond the basic : x = r cos theta y = r sin theta r = sq rt (x^2 + y^2) I totally don't get how to find the limits of integration using polar coordinates and my undergrad textbooks...
19. Quick Qns : Partial Derivatives

Suppose we are given : PV = nRT, where n and R are constants. We are told to find the partial derivative dP/dV. Am I allowed to do this : P = nRT/V Then differentiate this w.r.t. to V. I disregarded the fact that V = 0 makes the RHS undefined. # This question came from...
20. Quick Qns : Convergence in Polar Coordinates

Suppose we investigating the limit of a function on R^2 as (x,y) tend to (0,0). We convert the function into polar coordinates. Then "(x,y) tend to (0,0)" is equivalent to "r tend to 0"? Theta (the angle) does not matter?
21. Intermediate Value Theorem : Is this version correct?

I recently came across a version of the Intermediate Value Theorem in "Cracking the GRE" by Princeton Review. Intermediate Value Theorem : f is a real function continuous on a closed interval [a,b]. Let m be the absolute minimum of f on [a,b] and M be the absolute maximum. Then for...
22. Real Sequences : Can some terms be undefined?

The definition of a sequence of real numbers is : a function from N to R. What is the standard way to handle a sequence like (1/log n) where the first term is undefined? Do we instead write (1/log (n+1)) so that the first term is defined? Or leave the first term undefined? The definition...