Recent content by glacier302

Given the fact that X and Y are independent Cauchy random variables, I want to show that Z = X+Y is also a Cauchy random variable. I am given that X and Y are independent and identically distributed (both Cauchy), with density function f(x) = 1/(∏(1+x2)) . I also use the fact the...

Homework Statement Let X and Y be independent random variables each having the Cauchy density function f(x)=1/(∏(1+x2)), and let Z = X+Y. Show that Z also has a Cauchy density function. Homework Equations Density function for X and Y is f(x)=1/(∏(1+x2)) . Convolution integral =...

Maybe it would help if I provided more information about the problem: I'm working on a Gibbs sampler problem. For the regular Gibbs sampler, I used beta distributions to sample the probabilities p and q. I have a vector I that I'm updating at each iteration, which depends on the current...

Hello, I am working on a problem in which I first sample two unknown probabilities, p and q, from Beta distributions, and then I want to sample both of them at the same time from a multivariate Beta distribution. This is the code that I have for sampling p and q individually from Beta...

Never mind, I think I've got it. Since I want to solve the integral ∫g(x)dx from x=0 to x=a = ∫∫(1/t)f(t)dtdx, inner integral from t=x to t=a, outer integral from x=0 to x=a, the function is bounded by the curves t=x, t=a, x=0, and x=a. Another way of writing this is that the function is...

Homework Statement Let f be Lebesgue integrable on [0,a], and define g(x) = ∫(1/t)f(t)dt , lower integrand limit = x, upper integrand limit = a. Prove that g is integrable on [0,a], and that ∫f(x)dx = ∫g(x)dx . Homework Equations In the previous problem, I showed that if if f and g are...

Aha. Thank you! I knew that if a bounded function is Reimann integrable then the Reimann integral and the Lebesgue integral of the function are equal. However, I always forget that this can be extended to unbounded functions as long as the Reimann integral is finite. Thanks again!

Thank you for your help. One question: How I do know that the derivative of that function is integrable? If f' were bounded, then the fact that it is only discontinuous at x = 0 would make f' Reimann integrable, and Reimann integrability implies Lebesgue integrability for bounded functions. But...

What is an example of an absolutely continuous function on [a,b] whose derivative is unbounded? I know that the function f: [-1,1] defined by f(x) = x^2sin(1/x^2) for x ≠ 0, f(0) = 0 is continuous and its derivative f'(x) = 2xsin(1/x^2)-2/xcos(1/x^2) for x ≠ 0, f'(0) = 0 is unbounded on...

Homework Statement Prove that if f: [a,b] -> R is absolutely continuous, and E ∁ [a,b] has measure zero, then f(E) has measure zero. Homework Equations A function f: [a,b] -> R is absolutely continuous if for every ε > 0 there is an δ > 0 such that for every finite sequence {(xj,xj')}...

Homework Statement Let f:[a,b] -> R be a bounded function, and let D be the set of points at which f is not continuous. (a) Prove that D is a countable union of closed sets. (b) Prove that if m(D) = 0, then f is measurable. Homework Equations Of(x) = lim(ε->0)(sup f(y) - inf...

Homework Statement Let E be the set of points in [0,1) whose decimal expansion contains somewhere the block 55555. Find the measure of E. Homework Equations The Attempt at a Solution I have a feeling that in order to find the measure of E, I should find the measure of the...

I think I figured it out. Thank you!

Homework Statement Let An, n = 1,2,..., be a sequence of measurable sets. Let E = {x: x∈An i.o.}. (a) Prove that E is a measurable set. (b) Prove that m(E) = 0 if ∑m(An) < ∞ Homework Equations A point x is said to be in An infinitely often (i.o.) if there is an infinite...

Hello, Thank you very much for your help. I see now that using matrix exponentials is not the easiest way to solve this problem. Thanks again! : )