# Tagged Questions

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### Cesaro means converges in first mean to 0.

I would appreciate any suggestions to prove the following statement If $\{ X_i: i=1,..\}$ is a sequence of independent uniformly integrable (U.I) random variables ...
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### Soft Question: What are some elementary motivations of using functional analysis to study probability theory?

Recently i've become curious about the links between functional analysis and probability theory. What are some simple reasons why a functional analytic approach is preferable to a measure theoretic ...
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### Methods to distinguish continuous probability distributions

I read in the Wikipedia article for Variance The variance is one of several descriptors of a probability distribution. In particular, the variance is one of the moments of a distribution. In that ...
I am trying to show that if $X_t$ is some process and there is a function $p$ such that $$P[(X_{t_1},...,X_{t_n}) \in A_1 \times...\times A_n] = \int_{A_1 \times...\times A_n} ... 1answer 32 views ### American Put question If the interest rate is zero. Then show that the optimal exercise for an american put option is always the terminal time. That is, it is equivalent to a european put option. 1answer 65 views ### conditional probability of continuous independent random variables I would like to know what's the exact way to obtain the conditional probabilities of node having multiple independent continuous random variables as its parent. Say something like a Noisy OR gate ... 1answer 91 views ### a generalization of normal distribution to the complex case: complex integral over the real line How to prove \int_{\mathbb{R}} e^{-\frac{(x+it)^2}{2}}dx=\sqrt{2\pi} for any t\in \mathbb{R}? I only obtained the case that t=0, \int_{\mathbb{R}} e^{-\frac{x^2}{2}}dx=\sqrt{2\pi}. Thanks. 0answers 14 views ### Trouble with Largrange Multipliers and expectation. I am reading the following argument: Maximize E[Log(X(T))] subject to E[Z(T) X(T)]=x, where X(T),Z(T) are random variables, x is a constant, and E is expected value (You can read this as ... 0answers 24 views ### Understanding certain parts of the proof of Helly's Selection Theorem I have read through the following proof of Helly's Selection Theorem. There are just two parts, which I have highlighted, that are left for the reader to fill in, and I would like to know how to prove ... 1answer 66 views ### Is there closed form for (1-p)(1-p^2)(1-p^3)… or its Taylor expansion? I was considering the following problem: Somebody uses a backup for something (e.g. backups a file) and the backup is equally reliable as original storage. The storage is not perfectly reliable and ... 1answer 65 views ### Solutions to a stochastic birth-death-immigration process A population is undergoing a birth-death-immigration process. That is, the population size can increase by virtue of birth and immigration, and can decrease by virtue of death. The birth rate is ... 2answers 97 views ### Problem with infinite product measures Given some measurable space \left(X,\mathcal{F}\right) and two probability measures \mu and \nu on this space one can define ... 1answer 42 views ### solve the functional equation Let \phi : R-> C  (complex numbers) \phi(0)=1  \phi(-t) = \overline{\phi(t)}  ( continuous and bounded) solve the functional equation: Re \phi(t)= \phi(t) \overline{\phi(t)} This is all ... 1answer 70 views ### Show that if E is measurable set and f is continuous on E, then f(E) is measurable set Please tell me how to prove or disprove it ! Show that if E is measurable set and f is continuous on E, then f(E) is measurable set 1answer 56 views ### Convergence in Probability of a Sequence of Exponential Random Variables If X is an exponential random variable with \lambda = 3 and Y_n = \frac{X^n}{n}, I am trying to prove whether or not Y_n converges in probability. My original approach was the following: ... 0answers 41 views ### Convergence of eigenvectors/eigenvalues for infinite, non-negative, irreducible matrices with row sums less than 1 I actually had two questions but decided to put them in separate posts for clarity. Suppose you have a matrix M with infinitely many rows and columns, and a sequence of matrices M_m with the ... 1answer 40 views ### Properties of logarithmic mean. I have been studying the logarithmic mean for the last few days now. Could someone please help me with the following two questions? 1) We know that the log mean is in between the geometric mean and ... 0answers 32 views ### Linearization of exponential function with expectation I ran into the following formula in my class:$$ E[\exp(X)]\approx \text{constant}+E[X]. $$Since the linearization takes place at X=0, I don't understand why the constant in the formula is not ... 1answer 27 views ### Set Difference Probability [duplicate] Here is the question: Prove that for every \epsilon>0 and every set A\in\mathcal{B}(\mathbb{R}^{n}) there is a compact set K\subset A such that P(A\setminus K)\leq\epsilon. --I have ... 1answer 80 views ### The probability distribution function of uniform random variables is as given Given U_1, U_2, \dots, U_n where each U_i \sim U[0,1], then use uniqueness theorem to show probability distribution function of X = U_1 + U_2 + \ldots +U_n (sum of independent uniform random ... 1answer 46 views ### \mathbb L^1 + a.s. convergence of sequence (X_n) does not imply \sup(x_n) is integrable I'm looking for a counterexample. The setting is this: Given an probability space (\Omega,\mathbb{F},\mathbb{P}) , I look for sequence of random variables (X_n)_n and a random variable X, all in ... 2answers 83 views ### how to solve this integral in survival analysis Let T be a positive random variable, S(t)=P(T\geq t). Prove that$$E[T]=\int^\infty_0 S(t)dt.$$I have tried this unsuccessfully. 0answers 60 views ### A naive question about “random” probability distributions So I've come up with this idea, which may be mathematically unprecise, but it goes as follows. Example: Suppose you have a random variable X taking values in the interval [-1,1]. Then, say, ... 1answer 44 views ### Calculating the odds of winning a card game So my friend made this game and I want to the odds of winning his game. So his game is basically I pay \$$1$to draw$2$cards from the deck and guess$2$numbers and one suit. For each correct guess ... 1answer 125 views ### Infinitely Often vs Almost Always Let$A_n$be an event ($n \in \mathbb{N}$). Why is it that$x$occuring for all but finitely many$n$be a subset of$x$occuring for infinitely many$n$? What if$x \in A_{2k}$, then$x$will be ... 0answers 61 views ### Integral of the product of the Gamma function and the Confluent Hypergeometric function This question was posted previously on stats.SE. The characteristic function of the Fisher$(1,\alpha)$distribution is: $$C(t)=\frac{\Gamma \left(\frac{\alpha +1}{2}\right) ... 2answers 121 views ### Too stupid to understand random variable questions? I have two excercises: 1.) Let X_1,X_2,X_3 be independent uniformly distributed random variables on [0,1]. What is the density function of X_1+X_2+X_3? 2.) Let X_1,...,X_4 be independet ... 1answer 72 views ### Inequalities of the quantile function [closed] I'm trying to rigorously prove the following inequalities involving the quantile function Y(a) = \inf \{x \in\mathbb{R} : a \leq F(x)\} where F is the distribution function: 1) F(x) < a \iff ... 0answers 68 views ### Why does it exist? I can show that \lim_{a \rightarrow \infty} \int_{-\infty}^{a} (\cos(tu)-e^{-\frac{t^2}{2}})e^{-\frac{u^2}{2}}du converges to 0 but I am not sure why this implies the convergence of$$\lim_{b ... 2answers 138 views ### Probability that$\frac{n}{2}$bins are empty [close] A Bloom filter of length$n$was built. I have only the first$\frac{n}{2}$bits of this filter. How will the false positive probability change? For the whole Bloom filter, the false positive ... 0answers 45 views ### How much larger is the$\sigma$-algebra than the algebra in Caratheodory extension? Given a 'measure'$\lambda$on an algebra$\mathcal{A}$of sets, Caratheodory gives a way to extend this$\lambda$to a$\sigma$-algebra. The idea is we define an outer measure (on all subsets) ... 1answer 49 views ### IFF conditions for convergence in probability and almost surely I am working on a bunch of problems in preparation for an exam in Probability Theory. I have come across two similar questions that I need some assistance with. Suppose we have a sequence of ... 1answer 38 views ### Definition of a random variable in the context of a hypergeometric distribution We defined a random variable in a probability space$(\Omega, E, P)$as a map$X: \Omega \rightarrow \mathbb{R}\$. Unfortunately, I somehow have the impression that this term "random variable is used ...
I want to evaluate $$\frac{1}{\sqrt{2 \pi } \sigma}\int_{-\infty}^{\infty} x^2e^{-\frac{(x-\mu)^2}{2\sigma ^2}}dx.$$ The problem is, I don't want to run into heavy calculations. Therefore, maybe ...