0
votes
0answers
16 views

Hypercontractivity of Markov Operator

I have been reading a paper by Ahlswede and Gacs on hypercontractivity of Markov operator (see here 1) and its application to information theory. To be honest, I could not fully understand the ...
0
votes
0answers
12 views

Limiting distribution of loss random variable?

I'm going to try to make the notation not actuarial-specific, but for those with a background in actuarial science, this relates to exam MLC. Suppose I have random variables $X_{i} \geq 0$ such that ...
1
vote
0answers
31 views

Determining the Cramer-Rao lower bound

Let $X = (X_1,\dots,X_n)$ be a vector of iid variables from the smooth density $f(x,\theta_0), \theta_0 \in \Theta \subset \mathbb{R}$. Let $L(\theta)$ be the likelihood and $I(\theta)$ the ...
3
votes
2answers
81 views

How to give rigorous proofs of these two limit statements?

Let $X$ be a random variable with cumulative distribution function $F(x)$. Then how to rigorously prove the following two limit statements? $\lim_{x \to - \infty} F(x) = 0$. $\lim_{x \to + \infty} ...
4
votes
1answer
55 views

Uniform sampling with replacement item frequency

Suppose we are sampling from $N$ distinct items uniformly with replacement $M$ times. What can be said about the distribution of frequencies of items drawn? For example, if I sort all the frequencies ...
2
votes
0answers
42 views

Jacobian for a matrix transformation: Example of Cholesky decomposition

I would like to generally understand how the Jacobian of a matrix transformation can be computed. As a concrete example, consider the Transformation from a (correlation) matrix to its Cholesky ...
0
votes
0answers
50 views

Writing probability as log

I have a question regarding the log probability and I am confused on this. The question is: $$\hat P^{(t)}(x)=\sum_{i=1}^N v_i^{(t)}P_i^{(t)}(x)$$ which is some function of size $N$. The question ...
0
votes
1answer
31 views

How to check hypothesis in statistical data?

I have a statistical problem. In a city there are some hostels which differ by the number of rooms. The input data are the following. In a table there is information about hostels and corresponding ...
4
votes
1answer
142 views

Unbiased asymptotic variance

Problem: Let $X_1,...,X_n$ be indep. r.v.'s that satisfy, for $i = 1,...,n$, $E(X_i) = \mu_i(\theta)$ & $\mathrm{Var}(X_i)= \sigma_i^2(\theta)$. $\theta$ is the parameter of interest and the ...
1
vote
2answers
62 views

rationalwiki on “Extraordinary claims require extraordinary evidence”

I don't have a strong background in probability/statistics and I'm trying to understand the example at ...
0
votes
1answer
32 views

Probability for incomplete information

Let's say there are 10 teams: A-J. Only 1 team wins, others lose. Probability of any team to win is unknown (different for each team) and to be calculated. Not all teams participate in each game. ...
0
votes
1answer
35 views

How to calculate probability that a team will win

Let's say there are 10 teams: A-J. Each team always participate in each of the game. Only 1 team wins, others lose. Probability of any team to win is unknown (different for each team) and to be ...
2
votes
2answers
73 views

$X = (X_1, X_2)$ is it not a multivariate random variable?

$X=(X_1,X_2,\ldots, X_P)$ is a $p$-dimensional random variable on $(\Omega, S, P) $ iff $X_i$'s are univariate random variables on the same probability space $(\Omega, S, P)$ ." We all know ...
2
votes
1answer
66 views

Proving $E_{\theta}[T(X)] = \frac{\psi'(\theta)}{\eta'(\theta)}$

I'm trying to understand how to prove the following theorem: Let $\{P_{\theta}, \theta \in \Theta\}$ be a family of distributions in the one parameter exponential family with density (pmf) ...
-2
votes
0answers
32 views

What should the contestant do? [duplicate]

Suppose there are three curtains. Behind one curtain there is a nice prize while behind the other two there are worthless prizes. A contestant selects one curtain at random, and then one of the other ...
1
vote
1answer
21 views

How to make this bet fair?

A person bets $1$ dollar to $b$ dollars that he can draw two cards from an ordinary deck of cards without replacement and that they will be of the same suit. How to find the value of $b$ so that the ...
1
vote
1answer
21 views

How to compute of each player winning this sequence of games?

Players A and B play a sequence of independent games. Player A throws a die first and wins on a "six." If A fails, then player B throws and wins on a "five" or "six." If B fails, then A throws and ...
1
vote
3answers
39 views

How to compute these probabilities?

A pair of dice is cast until either the sum of seven or eight appears. How to compute the probability of a seven before an eight? Now, if this pair of dice is cast until a seven appears twice or ...
0
votes
2answers
47 views

How to compute this probability?

A drawer contains eight different pairs of socks. If six socks are drawn at random and without replacement, how to compute the probability that there is at least one matching pair among these six ...
0
votes
0answers
19 views

How to establish the independence or otherwise of these compound events?

Suppose that $C_1$, $C_2$, $\ldots$, $C_n$ are mutually independent events in a sample space $S$. Then how to establish the independence or otherwise of these combinations of events? $C_1^c$ and ...
0
votes
4answers
53 views

What is the probability that at least one letter is in the correct envelope?

A secretary types three letters and the three corresponding envelopes. In a hurry, he places at random one letter in each envelope. What is the probability that at least one letter is in the correct ...
1
vote
2answers
28 views

How many bulbs should be inspected for probability to exceed $1/2$?

In a lot of $50$ lightbulbs, there are $2$ bad bulbs. How many bulbs should be examined so that the probability of finding at least $1$ bad bulb is at least $1/2$? My effort: Suppose $n$, where $0 ...
1
vote
1answer
36 views

Confidence interval for estimating probability of a biased coin

Suppose we have a coin with a probability $p$ of coming up heads and $q = 1-p$ of coming up tails on any given toss. (A coin is biased unless $p=0.5$). But we are not given what $p$ or $q$ are. We ...
1
vote
0answers
7 views

Sampling with an “oversampling” factor, in K-Means||

I'm trying to understand K-Means||, a scalable version of K-Means++, which itself is an "improved" version of the clustering algorithm K-Means. Please find here the link to K-Means|| paper ...
0
votes
1answer
36 views

Variance of sum of multiplication of independent random variables

Suppose that we have $Z=\sum_{i=1}^n (a_i+b_iX_i)(c_i+d_iY_i)$. Where $a_i,b_i,c_i$ and $d_i$ are real numbers and $X_i$s and $Y_i$s are all independent random variables. How can I find the variance ...
0
votes
1answer
20 views

Median value resulting in negative number

I know the that the formula for finding the median of grouped data that is: $$\mathrm{Median} = L_m + \left [ \frac { \frac{n}{2} - F_{m-1} }{f_m} \right ] \times c$$ And I also know what the ...
0
votes
0answers
15 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
140 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
33 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
36 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
30 views

probability theory books [duplicate]

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
46 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
42 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
38 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 ...
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
50 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
42 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
34 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
20 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
44 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
32 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
32 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
85 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
18 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
45 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
45 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
40 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 $X$. ...