Tag Info

Hot answers tagged

2

You could require $f$ to only be continuous, but the idea is to have as few test functions as possible to satisfy the definition. Another way to say this is, the above possible values for $\{...\}$ are equivalent to the requirement that the equality holds for continuous functions. Here is a proof of that for the continuous bounded characterization. On one ...


2

I think this satisfies the requirements: $$X_n = \begin{cases} \dfrac{1}{n}\mu_0, & \text{with probability $\dfrac{n-1}{n}$} \\[2ex] \left(n-1+\dfrac{1}{n}\right)\mu_0, & \text{with probability $\dfrac{1}{n}$.} \end{cases}$$ We have $E(X_n)=\mu_0$ and $E(\sqrt{X_n})=\sqrt{\mu_0}\left(\dfrac{n-1}{n^{3/2}} +\dfrac{\sqrt{n-1+1/n}}{n} \right) ...


2

By definition: $$\begin{align} F_Z(z) & = \mathsf P(\min \{X,Y\}\leq z) \\[1ex] &= 1-\mathsf P(\min\{X,Y\}>z) \\[1ex] &= 1-\mathsf P(X>z)\mathsf P(Y>z) \\[1ex] & = 1-(1-F(z))^2 \\[1ex] & = 1 - (z+1)^{-4} \end{align}$$ You can take it from here.


2

The random variable is not limited to values less than $x$. For instance, I can show you that the function $$ X(t) = \frac{1}{t} $$ is a measurable function on $(0, 1)$. Here's how. Let's look at $$ \{ t \mid X(t) \le 11 \} $$ That's the set of all points in the domain for which $X(t) = 1/t$ is less than 11, which is exactly $$ A = \{t \mid \frac{1}{11} ...


2

A random variable $X$ on $\Omega$ is no more and no less than a function $X:\>\Omega\to{\mathbb R}$ satisfying the technical condition that it is measurable: For any $x\in{\mathbb R}$ the set $\{\omega\in\Omega\>|\>X(\omega)\leq x\}$ belongs to ${\cal F}$. This guarantees that for any two given values $a$, $b$ the probability $$P[a\leq X(\omega)\leq ...


1

The actual definition is that a function $X:\Omega\to\mathbb R$ is called a random variable when $X$ is $(\mathcal F, \mathcal B(\mathbb R))$-measurable. In other words, $X^{-1}(B)\in\mathcal F$ for any $B\in\mathcal B(\mathbb R)$, where $\mathcal B(\mathbb R)$ is the Borel $\sigma$-algebra on $\mathbb R$. Now, it is sufficient that ...


1

HINT: Write $\langle x_n:n\in\Bbb Z^+\rangle\sim\langle y_n:n\in\Bbb Z^+\rangle$ if and only if there is a finite permutation $\sigma$ such that $y_n=\sigma(x_n)$ for each $n\in\Bbb Z^+$. Check that $\sim$ is an equivalence relation on $S$. Show that an event is exchangeable if and only if it is a union of $\sim$-equivalence classes. Since the ...


1

While sample set could also be worthwhile terminology, it is worth recalling that any space in mathematics is always a set with some sort of operations defined. For example, in the theory of stochastic processes, we could let $\Omega = \mathcal{D}(\mathbb{R}_+, \mathbb{R})$, where $\mathcal{D}(\mathbb{R}_+, \mathbb{R})$ is the Skorokhod space of real-valued ...


1

You have been given that $$f_{X,Y}(x,y) = \begin{cases}4xy & : 0 \leq x \leq 1,\; 0 \leq y \leq 1\\ 0 & : \text{ elsewhere}\end{cases}$$ You wish to know where $f_{X,Y}(x, z-x)$ is supported (ie: not zero) with respect to $x$, for values of $z$ where $0\leq z\leq 2$.   (Since $z=x+y$, then $0+0\leq z\leq 1+1$.) $$f_{X,Y}(x,z-x) = ...


1

HINT If $X = \max(X_1,\ldots,X_n)$, then $X < c$ iff $X_i < c$ for all $i$. Probabilities multiply over independent events.



Only top voted, non community-wiki answers of a minimum length are eligible