0
votes
0answers
13 views

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 ...
6
votes
3answers
107 views

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 ...
0
votes
2answers
32 views

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 ...
1
vote
0answers
32 views

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 ...
0
votes
0answers
27 views

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 ...
0
votes
0answers
41 views

Special case of Kullback-Leibler additivity

I have three random variables $X,Y,Z$. If $(X,Z)$ are an independent pair and $(Y,Z)$ are an independent pair, then the additive property of the Kullback-Leibler divergence says $K(X,Z|Y,Z) = K(X|Y) ...
2
votes
2answers
34 views

Showing that Y has a uniform distribution if Y=F(X) where F is the cdf of X

Let X be a random variable with a continuous and strictly increasing c.d.f. function F (so that the quantile function F^−1 is well-defined). Define a new random variable Y by Y = F(X). Show that Y has a ...
0
votes
1answer
34 views

Method of moments for Beta $(\alpha_1,\alpha_2)$ distribution

I am trying to solve for the first two moments of a Beta$(\alpha_1,\alpha_2)$ distribution. We know that the first moment is equal to: $\mu_1 = \frac{\alpha_1}{\alpha_1+\alpha_2}$ and the second ...
-1
votes
1answer
24 views

Asymoptotic distribution of identically distributed random variables [closed]

$Y_1, Y_2, ..., Y_N$ are independent and identically distributed random variables with the distribution function $F := F_{Y_1}$ and $F'_n(y) = \frac{1}{n}\sum_{i=1}^{n}\mathbf{1}_{\{Y_i \leq x\}}$ as ...
2
votes
1answer
47 views

I need help understanding this proof about convergence in distribution

The proof says that we used the fact that $(1-\epsilon)^\frac{x}{\epsilon} \rightarrow e^{-x}$ Why is this so? How do I prove this? Also, why do we need the fact that $\lfloor x/p_n \rfloor - ...
3
votes
1answer
48 views

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 ...
1
vote
1answer
37 views

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. ...
2
votes
0answers
32 views

Rate of convergence of a martingale

I have a question related to convergence rates of martingales: Assume that there is a sequence of maximized likelihood ratios: $ \frac{f_{\hat{\theta}_{n}} \left ( Y_{1},Y_{2},\dots,Y_{n} \right ) ...
0
votes
0answers
19 views

The distribution of ratio of two shifted gamma

I am wondering if anyone can help me to find the ratio of this distribution. Assume $S$ and $T$ are independent, where $S\sim Gamma(n-1/2, 4(1+\rho)\sigma^2)$ $S\sim Gamma(n-1/2, ...
1
vote
1answer
40 views

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 ...
1
vote
1answer
26 views

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 ...
0
votes
0answers
27 views

Proof of Pearson's chi squared test

i was reading proof of this theorem on http://ocw.mit.edu/courses/mathematics/18-443-statistics-for-applications-fall-2003/lecture-notes/lec23.pdf They showed, that $\frac{v_j-np_j}{\sqrt{np_j}} ...
0
votes
1answer
61 views

A point in a circle is selected at random. Calculate probability that point is closer to centre than circumference

State any assumption(s) you make Well, I decided to draw a circle with a center at the origin of a Cartesian plane. It had radius r so it's coordinates on the axes were (0, r), etc. I then drew ...
0
votes
1answer
14 views

On a real line R points a,b are randomly selected such that -2<=a<=2 and 0<=b<=3. Find the probability that | a - b | > 1

Let's say that C is the set where |a-b|>1 So I suppose you could say plot it as coordinates where the x-axis (labelled a) is from [-2,2] and the y-axis (labelled b) is from [0,3]. Now |a-b| must be ...
3
votes
3answers
52 views

How to understand the variance formula?

How is the variance of Bernoulli distribution derived from the variance definition?
0
votes
2answers
30 views

Finding covariance between profit and quality

The quality $X$ of an item is uniformly distributed on the interval $[0,1]$ and the profit $Y$ is given by $Y = X^5$. Find the covariance between $X$ and $Y$ . Can someone interpret this question ...
1
vote
1answer
40 views

Mean of random sum of random variable

Suppose that we have $X_1, X_2, \ldots$ is a sequence of i.i.d random variables with $E(X_i)<+\infty$ and $N$ is a random variable taking values in $\{1,2,\ldots\}$, $N$ is independent with $X_1, ...
1
vote
1answer
44 views

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 ...
0
votes
2answers
39 views

“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 ...
3
votes
0answers
27 views

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 $X$. ...
1
vote
1answer
60 views

Probability of event in normal distribution

Let $X$ be a random variable that is normally distributed and $X_1,\ldots,X_n$ be (independet) copies of $X$, then we can estimate this probability by using a simple Monte-Carlo estimator: $p := P (X ...
0
votes
0answers
36 views

Urgent Find the CDF of U [duplicate]

I am having problems with Part 2. I know the upper limit of Y is x+u in the formula. But what about the limits of X ? Please help me !!
0
votes
1answer
28 views

Find marginal and conditional distributions [closed]

Consider the probabiility density function $f_{X_1, X_2}(x_1, x_2) = \left\{\begin{matrix}\frac{1}{8x_2} \exp\left\{ -\left( \frac{x_1}{2x_2} + \frac{x_2}{4}\right)\right\}, & x_1 > 0, ...
1
vote
0answers
73 views

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 ...
0
votes
0answers
43 views

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 ...
0
votes
1answer
10 views

Weighted variance of a small sample

I am trying to calculate the variance of a small sample. I have the data: ...
0
votes
1answer
49 views

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 ...
2
votes
1answer
47 views

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, ...
0
votes
1answer
42 views

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 $b\geq 0.$ ...
0
votes
0answers
18 views

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 $E\varepsilon_1 = 0,$ $E\varepsilon_1^2<\infty.$ ...
1
vote
1answer
44 views

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 ...
4
votes
3answers
49 views

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]. ...
2
votes
1answer
37 views

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 ...
2
votes
0answers
31 views

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 ...
0
votes
0answers
25 views

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 ...
0
votes
1answer
31 views

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 ...
0
votes
1answer
35 views

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 ...
0
votes
0answers
26 views

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 ...
0
votes
0answers
39 views

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 ...
-1
votes
2answers
51 views

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 ...
1
vote
1answer
15 views

Confidence interval for difference in means

I need to obtain a 95% confidence interval for the indifference in the mean score overall I have the following data which states subject (A-L) and then Test and Retest A B C D E F G H I J K ...
2
votes
5answers
239 views

Find the distribution of $X_1^2 + X_2^2$? [duplicate]

Let $X_1$ and $X_2$ are independent $N(0, \sigma^2)$ which means (mean = 0, variance = $\sigma^2$) random variables. What is the distribution of $X_1^2 + X_2^2$? My approach is that $X_1\sim N(0, ...
0
votes
1answer
18 views

Is the statement: $p\left(\left.y\right|h^{-1}\left(\varphi\right)\right)=p\left(\left.y\right|\varphi\right)$ correct?

Say I have a likelihood function $p\left(\left.y\right|\theta\right)$ and I make the reparameterization $\varphi=h\left(\theta\right)$ using the bijective function $h$ with inverse $h^{-1}$. Then it ...
1
vote
2answers
44 views

Maximum Likelihood Estimator for Uniform Distribution

Can somebody please explain this example to me. I am struggling to see why the likelihood is $\frac{1}{\theta^n}$ only if theta is greater than the maximum x. Furthermore, why is it the case that ...
0
votes
1answer
50 views

Puzzle for Applying the Definition to a t distribution

The coeffcient of variation (CV) for a sample of values $Y_1,\ldots, Y_n$ is defined by $$ CV = S/ \bar{Y}.$$ Let $Y_1,\ldots, Y_n$ be a random sample of size $10$ from a normal distribution with mean ...