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