# Tagged Questions

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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 ...
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### A question about the stability of a property of the normal distribution

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 $X$, $Y$ ...
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### 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 ...
If $X\sim \text{Exp}(X)$ then for all positive $a$ and $b$, $P(X>a+b\mid X>a)=P(X>b).$ So given independent random variables $X \sim \text{Exp}(\lambda)$, $Y \sim\text{Exp}(\mu)$ we would ...
### How can a $\sigma$-algebra be “treated” or computed? Example
My question is: I have a random variable $X:\Omega \rightarrow \mathbb{R}$, the $\sigma$-algebra generated by $X$ is: $\sigma(X) := \{X^{-1}(B), B\in \mathcal{B}(\mathbb{R})\}$. But, imagine now that ...