# Tagged Questions

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### L1 expected error for scale/translation in densities

This is an example given in the article about testability (Devroye and Lugosi 1999) (link: http://repositori.upf.edu/bitstream/handle/10230/1024/375.pdf?sequence=1 ) page 7. First I will introduce my ...
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### Birthday “Paradox” - another, different, version!

Background Many people are familiar with the so-called Birthday "Paradox" that, in a room of $23$ people, there is a better than $50/50$ chance that two of them will share the same birthday. In its ...
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### Total boundness of Lipschitz densities

In the article Almost Sure Testability of Classes of Densities by Devroye and Lugosi in 1999. They claim in Example 10 (page 9) that Lipschitz densities on [0,1] with Lipschitz constant bounded by ...
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### Properties of the essential supremum

In the article about testability of densities (Devroye and Lugosi 1999, page 4) (link: http://repositori.upf.edu/bitstream/handle/10230/1024/375.pdf?sequence=1 ). They use some properties of the ...
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### probability theory books

since I'm going to deepen my knowledge of math stats I was wondering what book I should start from...I need one that covers in the very fine details the following topics trasformations among ...
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### Upper Bound on Mutual Information

I am interested in an upper bound on mutual information that I have been encountering frequently in the statistics and probability literature. I have yet to see the "purest" form of the inequality, so ...
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### measure of dependence for copula

I have some question about the paper of Schweizer and Wolff (1981). The question concerns about the following bound $$\int_0^1\int_0^1|C(u,v)-uv|\,du\,dv\leq\frac{1}{12}$$ where $C$ is any copula. ...
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### Square root of Chi-square distribution tends to $N(0,1)$

The question requires to show that $\sqrt{2\chi^2_n}-\sqrt{2n}$ converges in distribution to $N(0,1)$ as $n \rightarrow \infty$, which I dont know how to proceed. The question also has a first part ...
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### What are the implications of the definition of limiting distribution?

Given a sequence of random variables $\{X_n\}$, if $$\lim_{n\to\infty}F_{X_n}(x)=F_X(x),\qquad\forall x\in C(F_X),$$ then we say $X$ is the limiting distribution of $\{X_n\}$. My question is: Under ...
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### Independent Events or Random Variables

First recall the following definition of independent random variables. Let $(X_t)_{t \in \mathcal T}$ be a set of random variables, where $\mathcal T$ is an arbitrary index set. Then $(X_t)$ is ...
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### “Show experimentally” that for large $N$, $X$ appears to be normally distributed.

I'm a bit confused about the following problem: Let $X$ be the random variable $$X = \frac{X_1+X_2+...+X_N}{\sqrt{N}}$$ where $X_k$ is the outcome from the $kth$ flip of a fair coin where heads ...
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### Asymptotic Bounds for the distribution of $f_n(X_n)$.

Let $\{X_n\}_{n \in \mathbb{N}}$ be a sequence of $\mathbb{R}^{k}$-valued random variables defined on some probability space $(\Omega, \mathcal{F}, \mathbb{P})$ converging almost surely to $X$. ...
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### Concentration inequality of weighted sum of random variables given a tail inequality

I tried to solve the exercise below which can be seen as a generalization of the Bernstein's concentration inequality. However, I have difficulty bounding the moment generating function of $Z$ (see ...
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### Poisson/ jump process distribution for process $z(t)=2t+B(t)+\sum_{k=0}^{X(t)} J_k$

For the process: $z(t)=2t+B(t)+\sum_{k=0}^{X(t)} J_k$, where $X(t)$ is a poisson process with paramater $\lambda$, and: $J_k$ are i.i.d . random variables (jumps). $B(t)$=brownian motion. I want to ...
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### Weighted variance of a small sample

I am trying to calculate the variance of a small sample. I have the data: ...
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### How to find the expectation value?

Suppose that an insurer has an exponential utility function $u(x)=−2e^{-2x}$. What is the minimum premium $P^{-}$ to be asked for a risk X? After solving this we reached the following, So,only ...
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### Given two uniformly distributed independent random variables what should the PDF of multiplication of them? [duplicate]

I am a probability noob and I was solving the following problem, but my answer doesn't match the book's. The length and width of panels used for interior doors(in inches) are denoted as Xand Y, ...
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### Does method of moments give consistent estimator?

Let $\{x_i\}$ be identically continuously distributed variables (not independent in general, let's say it can be a stationary AR(1) model). Define function $f_b$ depending on parameter $b\geq 0.$ ...
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### Does method of moments work here?

Consider autoregression AR(1): $$u_t=\beta u_{t-1}+ \varepsilon_t, \quad t \in \mathbb{Z}.$$ $\{\varepsilon_t\}$ - i.i.d. random variables with $E\varepsilon_1 = 0,$ $E\varepsilon_1^2<\infty.$ ...
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### convergence in probability of function of random variables

Suppose that $X_1, X_2, \ldots, X_n$ be a sequence of i.i.d random variables. If we have $E(|X_1|^k) <\infty$ for some $k>0$ and $f(x)$ is a bounded continuous function on $\mathbb{R}$. Is the ...
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### Joint density problem. Two uniform distributions

This is the problem: An insurer estimates that Smith's time until death is uniformly distributed on the interval [0,5], and Jone's time until death also uniformly distributed on the interval [0,10]. ...
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### What is the distribution of $x^TAx$ when $x$ is gaussian ($A$ may be not symmetric)

Suppose that $A \in \Re^{d \times d}$, $x \in \Re^d$ and each component of $x$ is independently sampled from $N(0,1)$. I wonder what is the distribution of $x^TAx$. To be more concrete, how will the ...
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### paramter estimation (maximum likelihood) of a mixture density

I have this mixture distribution $f(x) =w \cdot \mathcal{LN}(\mu_1,\sigma) + (1-w)\cdot \mathcal{LN}(\mu_2,\sigma)$ where $\mathcal{LN}(\mu,\sigma)$ is a lognormal distribution. I now have random ...
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### A measurement of “likelihood” of a roll of $n$ independent normal random variables

I'll start with a general question, and later, as I expect it is possible that the general case does not have a satisfying answer, I'll post my specific problem. I am trying to find some measurement ...
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### A question on Markov chain

Suppose for two random variables $X$ and $Y$ we have $X\perp\!\!\!\perp Y$ and also assume that three random variables $X$, $Y$ and $Z$ form the following Markov chain: $X\to Z\to Y$. Do these two ...
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### Function of random variables: the ratio of two bounded random variables

If x,y and z are continuous random variables. $$f_X(x)=2ax\exp(-ax^2+aR^2)\ \ \ R<x<\infty \\ f_Y(y)=\frac{2by\exp(-by^2)}{1-\exp(bR^2)} \ \ \ 0<y<R \\$$ where $R$,$a$ and $b$ are ...
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### Unbiased estimator for geometric distribution for $e^\theta$

I want to find unbiased estimator of $e^\theta$ for geometric distribution P(X = k) = $\theta (1-\theta)^{k-1}$. Sample consists of 1 element $X_1$. We can't use maximum likelihood here to find any ...
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### Distribution of the squared Euclidean distance between two discrete random vectors

Suppose I have random vectors $\mathbf{x}=[x_1,\ldots,x_n]$ and $\mathbf{y}=[y_1,\ldots,y_n]$ where the elements of $\mathbf{x}$ and $\mathbf{y}$ are the i.i.d. draws of a random variables with the ...
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### find the mean value of x if The probability distribution of a discrete random variable x is given

The probability distribution of a discrete random variable x is $$f (x)= \begin{pmatrix}3 \\ x \end{pmatrix} (1/4)^x (3/4)^{3-x}$$ Find the mean value of x. Construct a cumulative distribution ...