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Hint: Write $F_X(x) = E[I\{X \le x\}]$ and use Fubini's theorem.

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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 ...

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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 thanx$. 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.

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