Recent content by shoplifter

Sorry, the previous post doesn't seem to display equations correctly: I meant, I found the value of \mathbb{E}[e^{isX+itY}] to be \frac{1}{2\pi}\int_{-\infty}^\infty e^{isx + itx^2}e^{-x^2/2}dx.

For the answer (which is \mathbb{E}[e^{isX+itY}]), I am getting the following quantity raised to power n: \frac{1}{2\pi}\int_{-\infty}^\infty e^{isx + itx^2}e^{-x^2/2}dx Is this correct? Thanks.

so I get the characteristic function to be \mathbb{E}e^{(-ins^2/4t + in(A_1\sqrt{t} + s/2\sqrt{t})^2)}. I'm guessing we can take the first (constant) term out of the expectation, as \mathbb{E}(c) = c. But I don't see an immediate way to calculate the second term, because the integral is too...

yes, I apologize. Suppose A_1, ..., A_n are iid standard normal variables, and say X = A_1 + ... + A_n, and Y = A_1^2 + ... + A_n^2. Then what's the char. func. of the joint probability distribution of X and Y? Apologies again for not being clear before.

thank you for your detailed responses. However, the original question I am trying to solve does not say "X is normal given that Y is chi-squared". It says something like, okay, here are n identically distributed independent standard normal variables, and let X be their sum, and Y be their...

What exactly is a "joint characteristic function"? I want the characteristic function of the joint distribution of two (non-independent) probability distributions. I'll state the problem below for clarity. So my two distributions are the normal distribution with mean 0 and variance n, and the...

Let x_1, x_2, ..., x_n be identically distributed independent random variables, taking values in (1, 2). If y = x_1/(x_1 + ... + x_n), then what is the expectation of y?

well, since ab = pq, p divides ab. but (p, b) = 1 and so p must divide a [this is obvious, trust me. if you don't see it, break them up into prime factors and deal with them case by case]. the other direction is symmetric.

How do I find the stationary distribution of the Markov chain with the countable state space {0, 1, 2, ..., n, ...}, where each point, including 0, can either a. return to 0 with probability 1/2, or b. move to the right n -> n+1 with probability 1/2? Thanks.

any help please? :(

yes, and i realize that we want the integral to converge when we take the inverse transform. so in order to do that, I'm guessing the denominator has to have a power > 1, which is why we have that condition on a. so it will fail the second time, i guess. but i can't formalize my argument (and...

prove that f(x) = (1 + |x|)^{-a} is the Fourier transform of some integrable function on R, when a > 1. what happens when 0 < a <= 1? how about the function f(x) = 1/(log(|x|^2 + 2))?

oh i got the fact that it's galois over the third one -- from the polynomial x^2 - 2i*b^2. what about the last one?

no, it isn't, but it doesn't follow that it's not galois, right? I've worked for a while, and can't find explicit separable polynomials whose splitting fields are precisely those, but then what does that prove? i really don't see a way to disprove they're galois extensions. even if they are...

also, thinking about the polynomial (x + \beta + i\beta)(x + \beta - i\beta)(x - \beta + i\beta)(x - \beta - i\beta) doesn't help, because it gives me the information that \mathbb{Q}(\beta + i\beta) is galois over \mathbb{Q}(\sqrt{5}), but nothing else, and i don't see how this helps in the problem.