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    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...
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    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...
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    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) =...
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    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...
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    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...
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    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) =...
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    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...
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    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 =...
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    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)...
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    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]...
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    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...
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    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...
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    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...
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    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...
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    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...
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    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 ->...
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    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...
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    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...
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    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...
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    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?
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    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...
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    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...
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