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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Overlap between disks generated through a Poisson process

I have a homogeneous spatial Poisson process on $[0-d; 0-d]$ with an expected number of points equal to $n$. I use the points to generate disks of radius $r$ without any overlap constraints. I am ...
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26 views

Is the chance of a variable also a parameter for a probability distribution?

I'm new to statistics and I'm a bit confused about the concepts of 'chance of a variable' and 'parameters of a probability distribition'. Is chance also a parameter? And if so: can computing the ...
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5 views

Error in Joint Modelling in R

I have a dataset which has the following variables: sex (indicator), risk (indicator), and age. I've done cox models with these variables no problem and even basic joint models with them no problem. ...
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19 views

Estimate of mean weight?

An Argentinean Rancher, who raises miniature Llamas, wants to have an estimate of the mean weight of his herd. The herd could be regarded as being a Normal population, with variance 225 lbs. He ...
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21 views

What kind of permutation/combination is this?

I have 2 letters to choose from, and 3 positions I can place them in, where order matters. Lets say letters A and B: The possibilities, from an intuitive sense are: AAA, BAA, BBA, BAB, ABB, BBB, ...
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35 views

a problem regarding conditional probability and binomial distribution.

Die A has 4 red and 2 white faces whereas die B has 2 red and 4 white faces . A coin (fair) is tossed once . If it falls head , the game is carried on by throwing die A alone. If it falls tail die B ...
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27 views

If $\theta = (59.3\pm 1.2)^{\circ} $, find $\tan(\theta)$ and it´s uncertainty.

How can I find the uncertainty? I need to use this formula: The general formula for uncertanties is: $\sigma_q = \sigma_q (a,b,c,...)$ and $\mathrm{Cov}(a,b,c,...)=0$ $\,\,$then $$\sigma_q ^2 = ...
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38 views

How to quantify how much my data resembles a linear relationship?

I have a bunch of data points in Excel for different test subjects that are each represented by unique colors in the following graph: I wish to quantify in Excel what seems obvious to my eyes, that ...
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1answer
66 views

Questions on finding expected value and variance on a Poisson distribution

FProblem: A student walks along a real line and tries to get to the origin. Each step he makes is random ; the larger the intended step, the greater the variance is of that step. When the student is ...
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54 views

Question about normal approximation on roulette

Problem: At roulette, you bet a dollar on red $30$ times in a row. Each time you win a dollar with prob. $18/38$ and lose a dollar with prob. $20/38$. Find approximately the probability that after ...
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37 views

Conditional variance of sum of two correlated random variables

Let $\theta\sim\mathcal{N}(\bar{\theta},1/\tau_\theta)$ be a normally distributed random variable, $\varepsilon\sim\mathcal{N}(0,1/\tau_\varepsilon)$ be a normally distributed noise term independent ...
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44 views

Question about normal approximation

Question: A class has $20$ students, each student tosses a fair coin $300$ times and records the number of heads. Approximately what is the probability that no one gets exactly $150$ heads? (Use ...
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20 views

Find the uniformly most powerful unbiased test(UMPUT)

Let $(X_1,X_2,\ldots,X_n)$ be a random sample from uniform distribution on interval $(\theta_1, \theta_2)$. Find a uniformly most powerful unbiased test of size $\alpha$ for testing $H_0: ...
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1answer
20 views

Multidimensional Central Limit Theorem

Let $\mathbf{U}^n = (U_1^n,U_2^n,\cdots, U_k^n) $ converge in distribution to $N(\mathbf{0},\Sigma)$. My question is that if $V_{k+1} \sim N(0,1)$ and is independent of $\mathbf{U}^n$ then is it true ...
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13 views

Autoregressive process and average return time

I am back in my maths courses but I doesn't arrive to assess the average return time of an $\operatorname{AR}(1)$ process. Has someone any idea? And after for an $\operatorname{AR}(2)$? Suppose ...
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1answer
17 views

A question about Fisher–Neyman factorization theorem

$f_{\theta}(x)$, then $T$ is sufficient for $\theta$ if and only if nonnegative functions $g$ and $h$ can be found such that $f_{\theta} = h(x)g_{\theta}(T(x)) $ The statement is: if $F(t)$ is a ...
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1answer
33 views

Determining the efficiency of $2 \overline X$.

Let $X$ be a random variable with pdf $$ f\left(x;\theta\right) = \left\{ \begin{array}{lr} \frac{3\theta^3}{(x+\theta)^4} & \text{if } 0<x<\infty \text{ and } 0 < \theta ...
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39 views

Evaluate forecasting quality

A statistical model calculate the probability that some independent event occurs. These events either occur or not (binary outcome). I am looking to evaluate the quality of my forecasting. I looked ...
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26 views

Avoiding over/under fitting & validation set size

I've already posted this question on stats.stackexchange a couple days ago, maybe someone here can help. I'm using a PLS model associated to FT-NIR spectroscopy to predict some properties of ...
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2answers
51 views

Finding a transform of X

Question: Y~ Uniform (0,1) and given Y=y, random variable X is binomial with parameters n and y. Find the transform (moment generating function) of X. Does X have a special name? Attempt: I was ...
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2answers
26 views

the meaning of Confidence Interval

I am a bit confused how we interpret confidence intervals. For the formula $\bar{X}\pm z_{\large\frac{\alpha}{2}}\large\frac{\sigma}{\sqrt{n}}$ ,we substitute $\bar{x}_{obs}$ for $\bar{X}$, but ...
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14 views

What is the premium such that it is equal to the $90^{th}$ percentile of the distribution of total claims?

A company has a one-year group life policy that divides its employees into two classes as follows: Class, Probability of Death, Benefit, Number in Class, A, 0.01, ...
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41 views

What is the relationship or difference between MLE and EM algorithm?

I am trying to study EM algorithm and Maximum Likelihood Estimation. Somehow, they both sound the same to me but can't really say the difference. Maybe I don't really understand any of them. I have ...
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1answer
36 views

Finding standrad deviation $\sigma$

Carton of milk can be reserved fresh for $20$ days in average, $\frac13$ from the milk cartons can reserved fresh for $22$ days or more. Let's assume that the period of fresh is exponential ...
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30 views

Derive a hypothesis test Z

Suppose a random sample of size 10 is taken from the random variable X which has the normal distribution with unknown mean $\mu$ and variance 4. You are requested to test the hypothesis $H_0:\mu = 0$ ...
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what do we get if we compare fisher information of two distributions?

What do we get if we compare fisher information of two distributions belonging to different family of distributions with shape and scale parameters say we compare it for generalised exponential and ...
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3answers
51 views

Finding maximum of two variables

Given $X$ is uniform on $[0, 10]$. Let $$Y = \max(5, X).$$ Determine Var(Y). I'm familiar with how to find the variance of a uniform random variable, as well as the max of two random variables. ...
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2answers
32 views

Degree of freedom in statistic

I am always having difficulty understanding what degree of freedom really is. I know that when I post this question people are going to tell me to do some research myself but I did it and I am still ...
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1answer
28 views

Conditional expectation of insurance payment

I'm trying to solve the following problem: An insurance policy is written to cover a loss, X, where X has a uniform distribution on (0, 1000). At what level must a deductible be set in order for the ...
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1answer
15 views

Univariate transformation/change of variable with integration

I'm trying to solve the following problem: My thought process is to do a univariate change of variable, so that $Y = \frac{V}{100,000}$. Then we have $\frac{\delta}{\delta v}[\frac{V}{100,000}] = ...
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2answers
28 views

Normal distribution with sample

I'm trying to figure out the best approach to this problem. I would assume that I can use the Central Limit theorem first and then a binomial cdf: Chocolate is packaged into jars using a computerized ...
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15 views

How does one find the Fisher Information of a MLE?

The Statement of the Problem: Suppose that $Y_1, Y_2, \ldots, Y_n$ constitute a random sample of size $n$ from an exponential distribution with mean $\theta$. Find a $100(1-\alpha)\%$ confidence ...
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30 views

Probability distribution of data and likelihood in Bayesian and Frequentive Statistics

I have recently been studying Bayesian as well as Frequentive Statistics (mostly null hypothesis significance testing) and am confused as to the meaning of the distribution of the likelihood and ...
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31 views

what is distance between two random point in two non overlapping circles?

I have a large circle with radius R that two set poisson random points with different densities in inside circle distributed. there are small circles to radius Rc in this large circle that points ...
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2answers
24 views

Finding standard deviation given joint probabilities

I'm trying to find the standard deviation of $Z = X + Y$ given the following table: I'm getting $E[Z^2] = 36.31$ and $E[Z] = 5.45$, giving me a variance of $Var[X] = 36.31 - (5.45)^2 = 6.6075$ ...
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1answer
24 views

Literature on discriminant analysis

Can anyone suggest a good book on discriminant analysis - comprehensible and detailed? (Kendall and Stuart write about the subject too concisely.) Thanks in advance.
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25 views

How does sampling affect the distribution of frequencies of individual types?

Consider a population of size N in which individuals can be of x different types. Take a sample (with replacement) of size ...
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1answer
33 views

How to prove the convergence in probability?

$Y_n$, $n = 1, 2, ...$ is a sequence of nonnegative random variables with their means converging to $0$. Can we show $Y_n \rightarrow 0$ in probability? Can anyone give me a hint? Thanks!
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1answer
23 views

Error in ratio approximation

Not being very familiar with statistics I have been trying to figure this out for some time: Suppose that I am trying to determine the relative amounts of several animal species. To do this, I find ...
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15 views

Does this condition imply the Lindeberg condition?

Suppose $X,X_1, ...,X_n,...$ are i.i.d. random variables with mean $0$ and $P(|X| < 1) = 1$, does this condition imply the Lindeberg condition? If yes, how to prove it? Thanks so much!
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29 views

Calculating Conditional Probability (Maybe using Bayes Theorem)

Well, I've a bunch of variables, viz $Age$, $Gender$, etc.. Age can take: values from $1$ to $6$ (which are actually coded) Gender can take: $0$ and $1$ I know the $$P(A=1),...,P(A=6), ... P(G=0), ...
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214 views

Finding a hidden “heavy” subset of random variables.

Let $X_1,\dots, X_n$ be independent non-negative random variables (with finite expectation and variance), and $0 < m < n$ be a fixed integer such that there exists a subset $S\subseteq [n]$ of ...
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21 views

Calculate the arithmetic mean of a random variable

Suppose that the results of a measurement (of let's say the number of citizens of a town, which is modeled by the random variable $X$) are given as follows: Out of 4000 towns: a) 500 have ...
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10 views

Qualitative comparison of parameterized sigmoid function (logit)

Having the logistic function to describe a 2 Alternative Forced Choice experiment: $f(x) = \frac{1}{1+e^{(a+bx)}}$ which from the results, I'm getting different sigmoids with different steepness. So ...
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1answer
17 views

How does measurement error affect the standard error of regression coefficients?

How does inclusion of the measurement error in the model, as $$Y_i + \varDelta_i = bX_i + \varepsilon_i$$ affect the standard error of least square estimators $\hat{b}$ of coefficients $b$? If I ...
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12 views

How to determine the value of a t statistic in hypothesis testing?

First of all, sorry for the stupid question. I am aware the internet already has answers to this, but I really couldn't find any answer I can understand. Long story short, I'm supposed to test the ...
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1answer
36 views

How to propagate uncertainties in the dependent variable when doing linear regression?

Let's say I have an independent variable $\vec x$ and a dependent variable $\vec y$ and measurement errors on my dependent variable that I know to be $\delta y$. For the sake of simplicity, let's say ...
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19 views

Minimax of Negative Binomial

Studying for the quals and I am in a deep confusion. This is an exercise from Mathematical Statistics: A Decision Theoretic Approach by Thomas S. Ferguson. Let $X$ has density $$f(x|\theta)={r+x-1 ...
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1answer
66 views

PMF estimation: concentration inequalities for the $l_1$ and $l_\infty$ errors

Assume that you are given $n$ i.i.d samples $X_1, ..., X_n$ drawn from a discrete distribution $p = (p_1, ..., p_k)$. We would like to estimate $p$ using the empirical estimator \begin{equation} ...
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1answer
53 views

What does statistical significance literally mean? [closed]

I am getting confused when trying to determine the statistical significance of my results. For instance, I was attempting to make a conclusion of reliability of a model after 100000 trials depending ...