0
votes
2answers
6 views

Determining level of significance when hypothesis is an interval

For a normally distributed sample: σ = 60 Sample size = n = 12 Sample mean = x = 3450 Null Hypothesis = H_o ≠ 3500 Hypothesis = H_1 = 3500 I need to determine the smallest significance level at ...
0
votes
1answer
24 views

Transformation theorem, Cauchy distribution

I have derived the density for the ratio of two independent random variables,via the transformation formula. In this way: $V = X/Y $ and $ U = X $ inversion yields: $Y = U/V$ och $X =U$ , the ...
2
votes
0answers
28 views

How to scale “probabilities” to a given mean?

I have a set of scores $x_i$, $i=1,\ldots,N$ (mimicking probabilities, $0\le x_i\le 1$) and I want to transform them so that the result has a given mean $m$, while remaining in the interval $[0;1]$. ...
0
votes
0answers
25 views

Can someone help me with the following math question/dilemma?

I have a pool of objects that are randomly selected from a global object database. The objects certain numeric attributes: The objects from the pool are fed to users in real time Users will either ...
1
vote
1answer
28 views

How to show that $\Phi(1-x)^{-1} =O(\sqrt{\log{x^{-1}}})$

In the middle of some proof, I have faced an expression $\Phi^{-1}(1-x) =O(\sqrt{\log{x^{-1}}})$, where $\Phi(\cdot)^{-1}$ is a quantile function of the standard normal distribution and $x \in (0,1)$. ...
1
vote
1answer
34 views

Sum of truncated normal random variable and normal random variable

I'm wondering if there is a closed-form pdf of sum of "correlated" normal random variable and truncated normal random variable. I found a paper providing the pdf for "uncorrelated" case, but could ...
0
votes
2answers
42 views

Check if a given function is a probability density function [on hold]

Given $f(x)=\tfrac1{π(1+x^2)}, ~x\in(-\infty, \infty)$, is it true that $f$ is the probability density function of some continuous random variable?
0
votes
1answer
49 views

Fast way to inverse B'CB+D

$\mathbf {A = B'CB}$, where $\mathbf A$ is of dimension $n \times n$, $\mathbf C$ is m by m, positive definite and symmetric, $\mathbf B$ is of dimension $m \times n$, and $n >> m$. Inversion ...
3
votes
1answer
43 views

Statistical test for “too perfect” random number generator?

I am attempting to characterize some random number generator programs in a very simple way. Specifically, I'm rolling a simulated 6-sided die $3 \times 10^8$ times and keeping a count of how many ...
-1
votes
0answers
11 views

what are the joint distribution functions and copula? [on hold]

Let $U$ and $V$ be two independent uniform (0,1) random variables and let \begin{eqnarray*} R &=&\sqrt{\frac{U^{2}+V^{2}}{2}},\\ A &=&\frac{U+V}{2}, \\ G &=&\sqrt{UV}, \\ H ...
0
votes
1answer
16 views

Is there a way to use the Generalized Mean to find the “best” possible mean to use for a specific data set?

I've recently learned about the Generalized Mean as an abstraction of the many different means, includeing the Geometric, Arithmetic, and Harmonic means, as well as others. It is my understanding ...
0
votes
0answers
52 views

Share oranges evenly

There is a boy in a street sharing his oranges with people coming across to him on a street. He has 100 oranges in his basket. The number of people walking toward the boy are unknown and varies in ...
0
votes
1answer
44 views

Quite confused about continuous probability distribution

I'm self studying probabilities and statistics, now facing this problem. Use the random variable to represent the exact number of inches yesterday rained. Then the answer showed me a figure ...
0
votes
1answer
29 views

Linear regression as $\dim(\beta) \rightarrow \infty$

Consider the linear regression, $$ Y_i = X_i\beta + U_i \qquad E[X_i'U_i]=0 $$ where $X_i=(1,W_{i},W_{i}^2,..\ldots,W_i^K)$ and $\beta \in \mathbb{R}^{K+1}$. The joint distribution of $(X_i,Y_i)$ is ...
2
votes
2answers
56 views

Probability of Getting a “Perfect Score” in the Card Matching Game Concentration

A person is playing the card matching game concentration. There are 40 cards, 20 pairs total. All the cards are shuffled and placed at random face down. A turn consists of two moves and a move is ...
2
votes
2answers
33 views

Dice Rolling 4d10 with a twist

Suppose I roll two 10-sided dice, 1 die has numbers o, 10, 20, 30 etc to 90. The second die has numbers 0, 1 ,2 etc to 9. These dice are used to create a number from 1 to 100 - example: the first ...
-2
votes
0answers
27 views

Probability problem (reliability) [closed]

Assume that there are 30 cells per string. The designer is using metallic interconnects welded to the solar cells and reverse-voltage-blocking diodes connected to the ends of the cell strings by ...
0
votes
0answers
38 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) ...
1
vote
1answer
26 views

Distribution of reversed k-th order statistics

Let $X_1,...X_n$ be i.i.d. Let $Y_{(i)}$ the $i$-th order statistic of that sample. The distribution function of the order statistic is given by $$F_{Y_{(i)}}(y) = \sum_{k=i}^n \binom{n}{k} y^k ...
0
votes
0answers
18 views

Confidence size and coverage probability in a confidence set?

Let $\theta \in \Theta \subseteq \mathbb{R^d}$ be the parameter of interest and let $\theta_0$ be the true population parameter value. Let $n$ be the sample size. Let $CS_n$ be the confidence set ...
-1
votes
0answers
26 views

continuous bayes formula [closed]

A standard derivation with $\mu = 1.3$ and $r = 0.15$ defines the 1D-Position $p_1$ of an moving object at time $t_1$. At $t_2$ the new position is measured three times: $X_1 = 2.1$, $X_2 = 2.3$ and ...
0
votes
0answers
16 views

Maximum-Likelihood estimator [closed]

Given are $N$ Elements, labeled $1, ..., N$. The Elements are randomly shuffeled and then $n$ ($n < N$) Elements are choosen ($x_1, ..., x_n$). How do I found the Maximum-Likelihood estimator ...
0
votes
1answer
27 views

Mantel-Haenszel $\chi_1^2$ statistic

I was doing a particular example from the book Epidemiologic Research by Kleinbaum(example 15.6) and didn't understood some basic statistical aspect. ...
-2
votes
0answers
22 views

Combination of X1.y1 + x2, y2

Thanks in advance. Request to provide your support to solve the problem. I Have a set of X and Y combinations as follows: ...
0
votes
3answers
38 views

Question about normal approximation and variance

This isn't so much a question about getting a right answer as much as it's about understanding a mathematical concept, but I will give you the problem that spawned it: An analysis of data shows that ...
0
votes
1answer
36 views

Understand step in computing marginal distribution of restricted Boltzmann Distribution

Proof taken from http://image.diku.dk/igel/paper/AItRBM-proof.pdf (page 24) I understand everything up to and including: (1) $$p(\textbf{v}) = \frac{1}{Z}e^{\sum_{j=1}^mb_jv_j} \prod_{i=1}^n\sum ...
0
votes
1answer
32 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
vote
1answer
70 views

Expected value over many trials

I am a poker player and was talking to my friend about expected value. He claimed that if you play far enough above your bankroll, expected value can be negative, even if you have a skill edge. I ...
-1
votes
2answers
58 views

One of the Actuary Exam P question.

For Company A there is 60% chance that no claim is made during the coming year. If one or more claim are made, the goal claim amount is normally distributed with mean 10,000 and standard deviation ...
-1
votes
1answer
23 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
4answers
120 views

Is it true that $\mathbb E[{\frac{X}{Y}]}={\frac{\mathbb E[X]}{\mathbb E[Y]}}$?

If $X$ and $Y$ are both random variables, does it hold $$\mathbb E\left[\frac{X}{Y}\right]={\frac{\mathbb E[X]}{\mathbb E[Y]}}$$ ??
4
votes
1answer
63 views

My data is not normally distributed: what can I do to estimate a tail probability?

Continuing on from my earlier question, I'm attempting to analyse the data qualitatively. In the following plot, I make $10000$ samples where I count "the number of clashes". I plot $n$ vs. the ...
0
votes
0answers
34 views

Simple Variance approximation I don't get

I have $$ \log(\lambda_j -1) = c + \alpha_{j-1} $$ From here I know $$ \lambda_j -1 = \exp(c+\alpha_{j-1}) $$ Then, they say $$ Var(\hat \lambda_j)\approx\ Var(\hat \lambda_j - 1) \approx ...
2
votes
1answer
46 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 - ...
0
votes
2answers
37 views

Finding mean from die probability

Example 4.4.5: Suppose that there is a 6-sided die that is weighted in such a way that each time the die is rolled, the probabilities of rolling any of the numbers from 1 to 5 are all equal, ...
3
votes
1answer
47 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 ...
0
votes
1answer
17 views

Deriving marginal effects in multinomial logit model

For the multinomial logit model, it holds that: $$P[y_i=j]=\frac{\exp{\beta_{0,j} + \beta_1 x_{ij}}}{\sum_h \exp(\beta_{0,h} + \beta_1 x_{ih})}$$. Now my book states that the marginal effect is as ...
1
vote
1answer
36 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. ...
0
votes
1answer
11 views

Stats “AND” Elimination

What's the logical behind eliminating this AND in a solution? p(he lives to 80 GIVEN he lives to 50) = p(he lives to 80 AND he lives to 50)/p(he lives to 50) = p(he lives to 80)/p(he lives to 50) ...
1
vote
1answer
30 views

What can we say about correlation coefficients?

If we are looking at sales and inventory data with a correlation of 50%, what can I conclude? My intuition tells me that 50 % of the movement in one of the variables is attributed to the others ...
6
votes
4answers
538 views

Closed form equation for win percentage of two battling armies

I was pondering a battle mechanic for a board game that is similar to, but simpler than battling armies in Risk. Consider one army of size X and a second army of size Y. The battle occurs by ...
0
votes
0answers
49 views

Probability Question involving Probability Mass Function/Random Variables

Problem: When a paging system transmits a message, the probability that it will be received correctly by the appropriate pager is p. To ensure that the message is correctly received at least once, the ...
1
vote
0answers
30 views

Simple question about showing independence

How does one show that it is possible for a random variable $Z$ to be independent of $A$ but also not independent of $X$ where $X=1\{A>B\}?$ Under what circumstances can this be true, ignoring the ...
4
votes
2answers
38 views

Conditioning on information about the moments of a random variable is trivial

Say we have some random variable, $X$. Is it always trivial to condition on information about the moments of $X$? For example, suppose we know that $\mathbb{E}(X)$ is positive. But ...
0
votes
1answer
13 views

Density of k-th order statistic

I know that the k-th order statistic density for the uniform distribution is given by: $f_{(k)}(y) = \frac{N!}{(N-k)!(k-1)!} t^{k-1}(1-t)^{N-k}$. This is the density if the $N$ samples are sorted in ...
1
vote
2answers
28 views

Find probability in card question

Suppose that a test for extrasensory perception consists of naming (in any order) $3$ cards randomly drawn from a deck of $13$ cards. Find the probability that by chance alone, the person will ...
2
votes
0answers
31 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 ) ...
3
votes
0answers
26 views

Gamma distribution Norming constant for extreme minima

the norming constants for extreme maxima of Gamma distribution is known and is give in link.springer.com/article/10.1007/s10687-010-0125-3. I would like to know is there reference or paper that states ...
0
votes
0answers
35 views

Mathematical Probability and Statistics( all the math need)

I would like some suggestions about mathematical techniques and knowledge are required to understand and master 2nd year undergraduate probability and statistics. I am mature student with some ...
1
vote
1answer
13 views

How to calculate possible permutations with varying constraints

I have created a random email generator. The email is put together with the following format: (random first name) (random separator) (random last name) (random integer) @ (random domain) 150 ...