Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis.

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i have done partial of this problem, need help on this

Problem: Start with initial capital $ x$ and consider a continuous time game of betting $1$ on the outcome of the Brownian motion Wt started at x at time 0. Namely, Wt=x+Bt = fortune at time t, where ...
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63 views

Why do we like sticking random variables into their own distributions?

Let $X$ be a random variable taking values in the set $S$. It has some distribution $f(s)$. Often in statistics, we are interested in the real valued random variable $f(X)$. Here are some examples: ...
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1answer
16 views

Does the Information Gain algorithm favor a high-entropy attribute or a low-entropy one?

This might not be mutual to mathematics but it does relate to Information-Theory. My question is: Does the InformationGain algorithm, in Decision-Tree machine-learning, favor a high-entropy ...
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1answer
16 views

how is chi square test interpreted for cross tables?

Stats newbiew here. I understand that I use the chi square test to be able to calculate the probability that an observed value is due to chance against an expected value. I understand that it can be ...
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5 views

If this mean time is estimated to be in excess of 7 days, a new process will be implemented to reduce production costs.

a) You have just graduated with a post graduate degree in business and have obtained a position with a large manufacturing firm. The director of marketing has asked you to estimate the mean time ...
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8 views

The prime minister will accept an error of $5 million in the estimate of µ.

Government officials in Canberra have recently expressed concern regarding overruns on military contracts. These unplanned expenditures have been costing Australians millions of dollars every year. ...
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1answer
458 views

Determine whether ARMA(p,q) is stationary and/or invertible?

Determine whether an ARMA(p,q) process is stationary and invertible such that $y_t = \sum_{i=1}^{p} \phi_i y_{t-i} + \sum_{i=1}^{p} \theta_{i} \epsilon_{t-i}$ with the restriction that ...
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1answer
10 views

A Basic Question E-views

I ask a question about E-views. Is the P-value in the picture less than 0.05 or greater than 0.05? I'm confused because of the presence of the sign '<' in front of 0.10. Please help mee. Thank you. ...
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1answer
330 views

Rao-Cramer lower bound for Rician distribution

I derived ML estimator for Rician distributed data and I am trying to show Rao-Cramer lower bound of $\hat{A}$ estimator variance. $$f(x_k|A,\sigma) = ...
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1answer
13 views

How is the t-statistic computed in this context?

According to pg. 55 of Statistics in Plain English, the t-statistic for the sample mean is $$ t = \frac{\bar{x} - \mu}{s} $$ where $\bar{X}$ is the sample mean, $\mu$ is the population mean, and $s$ ...
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2answers
222 views

Is a Riemannian metric positive definite or positive semidefinite?

From Wikipedia The Fisher information matrix is a N x N positive semidefinite symmetric matrix, defining a Riemannian metric on the N-dimensional parameter space, But a Riemannian metric is ...
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2answers
899 views

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 ...
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138 views

How do I prove and expand Bayesian Networks?

Attempting to understand Exercise 20 (pdf page 44) in the paper: (Warning: large paper; small exercise) Bayesian Reasoning and Machine Learning The party animal problem corresponds to the ...
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1answer
66 views

Is there any difference between statistical learning and machine learning?

Straight to the point, I'm a math student and I have a course this year called Statistical Learning. From the description, the course contains: Large datasets analysis, regression, principal ...
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188 views

How to model mutual independence in Bayesian Networks?

It's well known that 3 random variables may be pairwise statistically independent but not mutually independent, for an illustration see: example pairwise vs. mutual relations. Can mutual ...
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1answer
63 views

probability calculation for bayesian network

I am studying Bayesian belief networks and in that I am struggling to understand how probabilities are calculated. I found this article here and the network is this: The associated probabilities ...
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1answer
648 views

bayesian networks for regression

Would it be possible to use bayesian network for regression and/or prediction? I understand that it is a tool one can use to compute probabilities, but I haven't found much material about possible ...
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0answers
155 views

Building Bayesian Networks, Causality and Cyclic Reasoning

I am studying Bayesian Statistics and I am trying to get a good understanding on Bayesian Networks, which seems to be vital in order to make something useful in Machine Learning. Most of the texts I ...
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1answer
45 views

Bayesian Network/ Number of parameters

Please consider the following Bayesian Network out of $Graphical Models in Applied Multivariate Statistics" by Joe Whittaker: Now the factorization property says that the joint probability ...
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42 views

Normal Distribution - What should I study to understand these questions [on hold]

I got these questions from someone. I am not expecting anyone to 'solve' these, just need an idea of which specific topics I should read up on to help the concerned person understand, as I have been ...
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20 views

How to find most correlated items?

[Complete noob here, apologies in advance] I have a table which contains a value for each other element in the table. I want to find out what the clusters are of related values (for instance, of ...
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25 views

Why does Correlation Coefficient concern about the mean of the vector?

$$r = \frac {\sum_{i=1}^n (X_i-\bar X)(Y_i-\bar Y)}{\sqrt{\sum_{i=1}^n(Xi-\bar X)^2} \sqrt{\sum_{i=1}^n(Y_i-\bar Y)^2}}$$ This is exactly the $\cos$ of degree of the angle between vector $X-\bar ...
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1answer
23 views

Correlation and Linear Regression

I'm tasked with this question but unable to proceed on. Q: Calculate the linear product moment correlation coefficient between x and m for these samples: $$ \Sigma x=205,\\ \Sigma m=1240, \\ \Sigma ...
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2answers
10 views

Show that $\frac{\alpha+y}{\alpha+n+\beta}\in (\frac{\alpha}{\alpha+\beta};\frac{y}{n})$

Suppose you assign a $Beta(\alpha,\beta)$ prior distribution for $\theta$, and the you observed $y$ heads out of $n$ spins. Show algebraically that your posterior mean of $\theta$ always lies ...
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1answer
7 views

Why is the degrees of freedom for a matched pairs $t$-test the number of pairs minus 1?

Cross-posted at Stats SE. It is eligible for bounty in 12 hours, but before I put a bounty on it, I would like to ask it here. I am used to knowing "degrees of freedom" as $n - r$, where you have the ...
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7 views

What is the minimal value of Cohen's kappa?

We are in the context of two classifiers, classifying $N \in \mathbb{N}$ datapoints into $z \in \mathbb{N}$ classes. Let $h_{ij}$ be the count of objects being classified to class $i$ by the first ...
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17 views
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Sufficient sample size for 1 out of X sets

I want to create a "game" that aims at supporting the decision making. The user has to consecutively select one out of two random items. Each item belongs to one of X sets. My question is how many ...
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2answers
565 views

Statistics and Probabilities- Distributions

A quality control engineer tests the quality of produced computers. Suppose that 5% of computers have defects and defects occur independently of each other. I need to find the probability that the ...
2
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1answer
17 views

Paired T-Tests vs Independent

The effectiveness of a training course is examined, and performance of each individual in a group is taken both before and after, and the differences are used in a paired T test. Would it be possible ...
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1answer
17 views

Posterior density for θ - Gamma(α, β)

I want to find the posterior density for $\theta$ with a prior Gamma distribution $(\alpha, \beta)$. $$ f(x\mid\theta) = \frac 1 2 \theta e^{-\theta |x|}, \quad -\infty < x < \infty $$ From ...
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1answer
16 views

Testing the Uniformly Most Powerful Test against the alternative

Hi I am working on the following problem A single observation $X$ is made from one of three densities listed below with parameter space $\Theta=\{0,1,2\}$. \begin{align*} ...
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13 views

Can distributions scale to any given time scale?

I am working with some colleagues to set up alerts based on call center data, and I feel that something is wrong with how it is currently being done, but I'm not sure what exactly, or how to explain ...
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1answer
39 views

Finding the mean age of a wedding guest list

Earlier I asked this question. So, similarly, imagine the same type of list with elements having a guest and probability of attending, but now each guest also has an age. How can we calculate the ...
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1answer
24 views

Comprehensive conditional probability question

Bob owns a bakery. It is located on a busy road with a lots of individuals walking by the bakery each day. On average, $\lambda$ individuals go into the bakery per day, the number of individuals ...
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2answers
28 views

Understanding independence. Sample space $S = \{1,2,3,\ldots,20\}$ and events $A_n = \{x \in S: x \: \text{is a multiple of} \:n\}$

The one in blue are the solution. $x$ is the multiple of $n$ which mean $A_2$ should equal to $4$ right? and $P(A_2)$ should equal to $1/5$ (As $2/20$). Why in the solution, $P(A_2) = 1/2$ ? ...
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2answers
22 views

Finding Type I error

Suppose the sample size $n=16$ is drawn from a normal distribution with mean $\mu$ and standard deviation $\sigma = 4$. Consider the testing hypothesis $H_o:\mu = 0$ vs $H_a:\mu \ne 0$. Let the ...
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Find the Fisher Information for 1/𝜃 [on hold]

I have found that $I(\theta) = 1/\theta^2$ So for $I(1/\theta)$, I am getting: $\theta^2 (1/\theta^4) = 1/\theta^2$ I feel this is incorrect, since $I(\theta) = I(1/\theta)$. Can somebody confirm? ...
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1answer
14 views

Weighted Average or Non-Weighted Average for this?

I surveyed people for a thesis I made, and want to calculate the standard deviation of their results, however I am not sure whether or not I should use weighted average or non-weighted average, my ...
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7 views

Placing upper bound on an error rate. Contrasts. Stats.

I need some help understand what this actually is. My textbook asks a question but never discusses what placing an upper bound is. So a research study conducted an experiment with 5 treatments, four ...
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1answer
9 views

Calculating the ratio of services to total jobs installed in one year.

I have had a total of 69 jobs this year, 13 of my jobs have had services this year. What is the ratio of services to total jobs?
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51 views

Probability of number of people who know a rumor

Suppose that among a group of $n$ people, some unknown number of people $K$ know a rumor. If someone knows the rumor, there is a probability $p$ that they will tell it to us if we ask. If they don't ...
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3 views

How to determine the spectral distribution and its density for a stationary GARCH process?

I am having problems with determining the spectral distribution and its density for a stationary GARCH process. I have no idea what to do. I would appreciate some hints, not the solution but hints or ...
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1answer
24 views

Finding Uniformly Most Powerful(UMP) tests of size $\alpha$

Hi I am working on the following problem: Let $X_1,X_2,\ldots,X_n$ be a random sample from a distribution with PDF given by ...
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19 views

How is this not a counter-example to the fact that the standard deviation of a standardized random variable must be 1?

Let $X$ be a random variable that sends all members of the sample space to some constant real number $c$. Let $Z$ be its standardized counter-part. The standard deviation of $X$ is $0$, since $X$ is ...
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16 views

How to plot the distribution of a function in r? [on hold]

I want to plot the distribution of $\bar{X} = {X_1+X_2+\cdots+X_n\over n}$. How do I do this in R?
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8 views

Logistic regression probabilities

I have performed a logistic regression on some data, and the function is: $p(x,y) = \frac{\exp(4.5 + 3.5x - 0.1y)}{1+\exp(4.5 + 3.5x - 0.1y)}$ This also gives you the probability of observing some ...
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13 views

How to do non-analog sampling from a pdf

I have a distribution given by $$ f(x)=\frac{1}{2}\sinh(\sqrt{2x})e^{-x}. $$ Due to the shape of this distribution, most of my samples will be in the range $x\in(0,2]$. However, I am interested ...
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21 views

Finding the joint distribution and covariance matrix of a function.

Question: Let X and Y be two continuous random variables with joint probability density function $$f(x,y)=\begin{cases}\frac{1}{2} & \text{if} \ \lvert x \rvert + \lvert y \rvert \le 1 & \\ 0 ...
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9 views

Expectation Maximization (EM) for 3-dimensional parameter $(\alpha,\mu_2,{\sigma}^2)$.

Let $x_i$ where $i=1,...,100$ are iid observations from a mix of two normal distributions with means $\mu_1=0$ and $\mu_2$ and the same variance ${\sigma}^2$. If $\alpha$ is the proportion of the ...
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3answers
28 views

What is the difference between linear regression on y with x and x with y

I'm plotting the regression line of (GDP$\%$ Change, Poverty Rate$\%$)$\to (x,y)$ in Mathematica What would it mean if I were to switch the axis? (Poverty Rate $\%$, GDP change $%$) ...