# probability-theory's questions - Chinese 1answer

22.869 probability-theory questions.

### 1 Probability of an event $A$

1 answers, 18 views probability probability-theory
Let $\mathcal{Z}\subset \mathbb{R}$ be a set at most countable. Let $X,Y,Z,\xi$ random variables with the following properties: $X,Y,Z$ have value in $\mathcal{Z}$ $\xi\$ has a Bernoulli ...

### Discussing the set of solutions of a system of non-linear equations involving a parametric multivariate probability distribution

I want to study the set of solutions of a system of non-linear equations involving a parametric multivariate probability distribution. I would like your help to formalise or contradict some of my ...

### Associate a pvalue to set intersection

I have a set $S$ of $N$ elements; then I do three independent samplings from $S$, and I create three sets $R_1$, $R_2$ and $R_3$ of sizes $N_1$, $N_2$ and $N_3$. Let $N_{int}$ be the size of the ...

### 5 A Characterization of the Mode of a Distribution

Let $f:\mathbb{R} \rightarrow [0,\infty)$ be a continuous probability density function on $\mathbb{R}$ such that $$\int_{\mathbb{R}} |x| f(x)\, dx < \infty,$$ and ...

### 1 Sum of weighted chi square distributions

Let $X_1 \sim \chi_{k}^2$ and $X_2 \sim \chi_{k}^2$ be i.i.d and both $a_1$ and $a_2$ positive real values. How can be expressed the PDF of $Y = a_1X_1 + a_2X_2$? Is it also a chi-square distribution?...

### 8 Relative entropy for martingale measures

I need some help understanding a note given in a lot of papers I read. Let $(\Omega,\mathcal{F},P)$ be a complete probability Space, $\mathbb{F} = (\mathcal{F}_t)_{t\in[0,T]}$ a given filtration with ...

### 1 Representing random variables in a shared and private decomposition

I'm struggling with proving/disproving the following statement. Let $X$ and $Y$ be two random variables such that $Y \neq g(X)$ and $X \neq g(Y)$ for any function $g$. Then, there are random ...

### Comparing event $X$ and $Y$, if $P(X) = o(1)$ and $P(Y) = \Omega(1)$?

Consider two probabilistic events $X$ and $Y$: if $P(X) = o(1)$ and $P(Y) = \Omega(1)$ (where $o(.)$ and $\Omega(.)$ are little Oh-notation), is it possible to compare the two event $X$ and $Y$? (...

### If $X = g(Y) + Z$, with $Y,Z$ independent continu0us RVs, what is the conditional density $f_{X \mid Y=y}(x)$?

1 answers, 25 views probability probability-theory

### 1 Do independent random variables implies conditionally independence too?

0 answers, 40 views probability-theory soft-question
Let $X$ and $Y$ be mutually independent random variables i-e $$P(X,Y)=P(X) \cdot P(Y)$$ where $P(X)$ is the probability mass function of $X$. Is it possible that some condition (if exists) will make ...

### 1 A problem of conditional probability

Consider three events $A,B,C$ such that $P(A)>0$, $P(B)>0$, and $P(C)>0$. The events are linked to each other through the constraints $P(A\cup B\cup C)=1$ and $P(A)=P(\overline{B})$. Under ...

### 4 A weird problem of probability!

Consider three events $A,B,C$ such that $P(A)>0$, $P(B)>0$, and $P(C)>0$. The events are linked to each other through the constraints $P(A\cup B\cup C)=1$ and $P(A)=P(\overline{B})$. We ...

### Arithmetic Combination of Two Distributions and Chernoff Bounds

I have a set of i.i.d random variables $X_i$ with distribution: \begin{align} X\stackrel{d}{=} \begin{cases} \mathcal U[-z,z] & \text{ if } K_i \leq 0.5 \\ \mathcal U[-1,1] & \text{ if } K_i &...

### 52 Reference book on measure theory

I post this question with some personal specifications. I hope it does not overlap with old posted questions. Recently I strongly feel that I have to review the knowledge of measure theory for the ...

### 2 Rao-Blackwell Theorem and UMVUEs

If I use the Rao-Blackwell theorem to find that a conditional statistic has the same variance as the original statistic I conditioned on, does that imply that this statistic is a uniformly minimum ...

### 3 Collection of standard facts about convergence of random variables

The goal of this question is to collect standard general facts about convergence of random variables (in $\mathbb L^p$, in probability, in distribution) in order to use them when answering questions. ...

### 19 Applications of information geometry to the natural sciences

I am contemplating undergraduate thesis topics, and am searching for a topic that combines my favorite areas of analysis, differential geometry, graph theory, and probability, and that also has (...