# Tagged Questions

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### Integrating a function wrt different measures [duplicate]

Suppose that $(\Omega, \mathcal E, P)$ is a probability space and $X\colon \Omega \to \mathbb R$ is a RV defined on $\Omega$. Denote as $\mu\colon \mathcal B \to [0,1]$ the probability measure on ...
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### Integration of a continuous function under Lebesgue-Stieltjes measure space using simple functions

I am struggling to prove the following result using an approximating sequence of simple functions. Could anyone give me a clue? Under a Lebesgue-Stieltjes measure space ...
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### Ito integrals and joint distribution with copulas

Let $X_{t}$ and $Y_{t}$ be two brownian motions and let their joint distribution be given by $F$. So in regularly correlated BM's where $dX_{t}dY_{t}=\rho dt$, we have a bivariate normal distribution ...
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### Sum of random variable

Considering two continuous random variables $X$ and $Y$ with $d.f \; F_X, F_Y$ I want to fin the distribution and distribution function of the sum $Z=X+Y$. \begin{align} P\{Z \leq z\} &= P\{X+Y ...
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### Existence of a measure with given marginals on product space

Let $X_1,...,X_n$, $n\geq 2$ be Polish spaces. I have a given compatible family of probability measures $\{\pi_{ij} \in X_i\times X_j \}$ (here each measure is defined on the space of the form ...
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### Show that for $t\in\mathbb{R}$ it is $\int_{-\infty}^{\infty}(G(x+t)-G(x))\, dx=t$ (distribution function)

Let $G$ be the distribution function of a probability measure on $(\mathbb{R},\mathcal{B})$. Show that for $t\in\mathbb{R}$ it is $$\int_{-\infty}^{\infty}(G(x+t)-G(x))\, dx=t.$$ ...
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### Upper Bound on Supremum of Expected Value

Let $\left( \Omega, F, P\right)$ be a probability space, where $P$ is a probability measure on $\mathbb{X} \subseteq \mathbb{R}^n$, so that $P(\mathbb{X}) = 1$. For all integer $i \geq 1$, consider ...
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### How do you find the distribution of this sum?

If $X\sim \text{Normal}(\mu=0, \sigma^2), Y\sim \text{Unif}(0,\pi)$, and $X \perp Y$, how do you find the distribution of $Z=X+a\cdot cos(Y)$ for some $a > 0$ ? I've found the distribution ...
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### Probability of a random variable dependent on a parameter.

Let $X_L$ be a random variable dependent on a parameter $L$, taking only discrete values between $0$ and $+\infty$. Let $\mu L$ be its expectation, where $\mu$ is a costant. Which conditions should I ...
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### Expectation of composition of functions with density as R-N derivative

In prior probability courses, I've always seen and used the fact that, for a continuous random variable X and a function $\phi$, $E[\phi(X)]=\int_{ \mathbb{R}}\phi(x) f_X(x)dx,$ but I cannot find a ...
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### What is the measure of all probability distributions with finite variance?

I'm in over my head here, but I am wondering about the probability that a distribution has finite variance? (or a finite mean?) By this, I don't mean that there is some set of data, just over the set ...
Recall that the standard normal distribution can be characterized as the unique standardized (having mean zero and unit variance) distribution $P$ on $\mathbb{R}$ with the property that with $X$, $Y$ ...
### Conditional Expectation of Exponential Order Statistic $\text{E}(X_{(2)} \mid X_{(1)}=r_1)$
Having already worked out the distributions of $\Delta_{(2)}X=X_{(2)}-X_{(1)}\sim\text{Exp}(\lambda)$ and of $\Delta_{(1)}X=X_{(1)}\sim\text{Exp}(2\lambda)$ where $X_{(i)}$ are the $i$th order ...