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Questions tagged [statistics]

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

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How to understand a comparison in Huff's How to Lie with Statistics

In the book How to Lie with Statistics (pp. 56-57), Huff (1954) gave an example of comparing IQ scores to illustrate the significance of probable errors. [...] So Peter's indicated IQ might be more ...
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5 views

relationship between hypothesis test for slope of simple linear regression and r^2

As I understand it, when you fail to reject the null hypothesis that slope=0 for simple linear regression using a t-test - it would suggest that the linear model degenerates into an intercept only ...
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1answer
42 views

Interview Question on Probability: A and B toss a dice with 1 to n faces in an alternative way

A and B toss a dice with 1 to n faces in an alternative way, the game is over when a face shows up with point less than the previous toss and that person loses. What is the probability of the first ...
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1answer
35 views

When are geometric and harmonic means used?

From what little statistics I know, the only 'mean' commonly used is the arithmetic mean, and the rest are irrelevant. Any reading I've done has pretty much said something along the lines of "...
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SD for differences in mean of paired (dependent) samples

I am trying to construct confidence interval for paired data, n=10. Since each pair is dependent, the confidence interval has the following form: $$ \bar{D} \pm t_{n-1,\alpha/2}S_{\bar{D}}$$ ...
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2answers
39 views

Given density of X, find density of X^2 [on hold]

If $X$ is a normal random variable with mean $0$ and variance $1$, how would we find the density function of $X^2$
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1answer
29 views

Examples non experiments in probability

I'm reading a book on probability theory and they say that an experiment is Any procedure that has not got a pre-determined outcome In the Wikipedia page they, instead, that an experiment should: ...
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33 views

Addition of all probability units greater than actual probability [on hold]

As shown, X is the sum of all probabilities, Y is the exact probability of occurrence. Can I show $X$ is always greater or equal to $Y$? $X= \sum _{ \left( i,j \right) \epsilon A}^{}x_{ij}$ $Y=1- \...
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Find the distribution of $X_t|Y_t,X_{t-1}$

Find the distribution of $X_t|Y_t,X_{t-1}$ given $$X_t=\alpha X_{t-1}+V_t$$ $$Y_t=X_t+\sigma_wW_t$$ where $X_1 \sim N(0,1)$, $V_t \stackrel{iid}{\sim} N(0,1)$, and $W_t \stackrel{iid}{\sim} N(0,1)$. ...
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19 views

Why does the formula for the uncertainty due to bias contain the square-root of 3?

I have been tasked with using standard gage blocks to calculate the uncertainty of a super micrometer and must consider the uncertainty due to bias in my calculation. The formula I am using for that ...
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78 views

Showing $E(S^2\mid \bar X)=\bar X$ for i.i.d Poisson random variables $X_i$

Let $X_1,X_2,\ldots,X_n$ be i.i.d $\text{P}(\lambda)$ random variables where $\lambda(>0)$ is unknown. Define $$\bar X=\frac{1}{n}\sum_{i=1}^n X_i\qquad,\qquad S^2=\frac{1}{n-1}\sum_{i=1}^n(X_i-\...
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Standard measures derived from autocorrelation functions

Setting: Let $(x_i)_{i=1}^N$ be a discrete signal of length $N$. This question is concerned with discrete finite data sets for simplicity, but they can also be raised in more general contexts. The ...
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10 views

Plotting a posterior density in R

The prior distribution is $N(0,1)$ The likelihood is $X=(V-2)^{1/(2q+1)}$, where $V$ has a distribution $N(0,0.05^2)$. I calculated the posterior density to be $$f(t)\propto \exp\left(-\frac{(t^{2q+...
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3answers
44 views

Sampling from product of exponential distributions

I have a distribution who's moment generating function is the product of two exponentially distributed variables moment generating functions. If I wanted to generate samples from the distribution, ...
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1answer
21 views

Trouble finding inverse of CDF to sample from

I am attempting to do a discrete approximation using subintervals of a PDF's 99% CI. I understand that to generate samples from this distribution to use in my program, one can take the inverse of the ...
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1answer
68 views

What is “$\ldots$” called in English when it comes to counting rule for compound events?

Hi! How do you read this theory in English, mainly GenAm/RP? Theorem one point two. If an operation consists of k steps, of which the first can be done in n-subscript-one ways, for each of ...
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Is there a way to associate hurst values with possible switching points in time series?

I thought about using a local min and max to see if there was any association with turning points in trend and the hurst max and min, but that is really my only idea as of now. Any suggestions on ...
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1answer
17 views

Using Weights to Create a Ranking where the Weights can be negative

I've got a question that's a variation of this one: How to weight three different variables to create a ranking I'm starting a new question because of the age of the other question. The difference ...
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15 views

Choosing regression variables to minimize variance of coefficients

We have the following exercise to solve: A researcher wants to find out how variable Y is dependents on amounts of variable X, with n = 10 samples given. The i-th sample has an amount $x_i$ of X, ...
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1answer
60 views

Condition number of $AA^T$ when $A$ is polynomial Vandermonde

Suppose I'm doing polynomial regression of degree $m$ $$p(x, \mathbf{w}) = w_0 + w_1x + \dotsb + w_mx^m$$ given training data $(x_1, t_1), \dotsc, (x_N, t_N)$. Suppose I'm using the loss function $...
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1answer
18 views

Sequentially updating the variance (or higher order moments) of a sample (online learning)

In online learning, new data become available to us one after another. Computationally, it is desirable to have a method of learning about the distribution of the data without redoing all the ...
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1answer
42 views

Beta distribution as a member of the exponential family

I come across the beta distribution quite frequently when solving exercises for my statistics class. However, I have not been able to fully grasp how to work with it. Exponential family form is: $$...
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29 views

Null hypothesis without significance level

"In a factory we produce rods the length of which is planned to be $80$ cm. We assume that the length of the rods varies randomly having the normal distribution. A sample of $100$ rods is taken in ...
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2answers
27 views

Calculate the value given several percentiles and their values

On a particular test whose scores are distributed normally, the 2nd percentile is 1,720, while the 84th percentile is 1,990. What score, rounded to the nearest 10, most closely corresponds to the 16th ...
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1answer
38 views

Gamma distribution family and sufficient statistic

Let X $=X_1,...X_n$ be a random sample iid from the probability density function: $$ f(x;\theta)=\frac{\Gamma(\theta)\sin(\pi\theta)\theta^{1-\theta}}{\pi}e^{-\theta x}x^{-\theta}$$ $x>0, 0<\...
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2answers
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A Problem dealing with the Binomial Distribution

Below is my solution to a problem from a text book. I do not have confidence that my solution is right. I feel like I am missing something. Am I? Thanks, Bob Problem: An airline finds that $5$ percent ...
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2answers
24 views

ML estimation with given samples

Let $X_i,...,X_n$ be a random independent sample from a distribution with pdf $$ f(x;\theta)= (\theta + 1)x^{-(\theta+2)},$$ where $x>0$, and $\theta > 0$. What is the ML estimate for the ...
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1answer
29 views

derive sufficient statistic from a random independent sample from a weibull distribution

Suppose $X_i$ is a random independent sample from a Weibull distribution $$ f(x) = \frac{\beta}{\theta^\beta}x^{\beta-1}\exp\big(-(\frac{x}{\theta})^\beta\big)$$ Find a sufficient statistic for ...
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1answer
20 views

derive asymptotic distribution of the ML estimator

Let $x$ be a random variable with probability density (pdf) $$f(x)= (\theta +1)x^\theta $$ where $\theta >-1$. The expressions for its mean and variance are $$E(X)= \frac{\theta + 1}{\theta +...
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1answer
27 views

Chebyshev's inequality application and convergence - practical example

Let $W_n$ be a random variable with mean $\mu$ and variance $\frac{b^2}{n^{2p}}$, with $p>0$ and $b$ and $\mu$ constants. Show that $$ \lim_{n\to\infty} P(|W_n-\mu| \leq \epsilon) = 1 $$ ...
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1answer
25 views

Calculating expectation w.r.t. the empirical dist. fcn

I am trying to evaluate the plug-in estimator $$\hat{\theta} = T(F_n)$$ where $F_n (t) = \frac{1}{n} \sum_{i = 1}^{n}1\{t_i \leq t\}$ and $T : \eta \to \int_{}^{}x \ d\eta(x) $, (Riemann-Stieltjes ...
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1answer
20 views

Probability of exactly one event out of N

I've been all the morning searching for this and I ended with nothing clear... So I'm studying for my statistics exam an one excercise asks this: If we have four computers A, B, C and D, whith each ...
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0answers
28 views

Detection of certain stochastic process

Hello math stack exchange! Suppose we have arbitrary function of stochastic processes $X_i(t)$, like $$Y(t) = X_1(t) + X_2(t) - X_3(t) + X_4(t) ....,$$ more generally $Y(t) = f(X_1(t), X_2(t), X_3(t)...
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2answers
38 views

MLE of continuous uniform distribution

A series of $n$ geomagnetic readings are taken from a meter, but the readings are judged to be approximate and unreliable. The chief scientist does know however that the true values are all positive ...
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1answer
41 views

Show $T=X_1(X_3+X_4)+X_2$ is not sufficient for $p$ where $X_1,…,X_4$ are iid Bernoulli(p)

I am attempting to show that $T=X_1(X_3+X_4)+X_2$ is not sufficient for $p$ where $X_1,...,X_4$ are iid Bernoulli(p), and the question specifies that I am to use the conditional distribution method. ...
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1answer
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Add component to equation so a value turns into a negative?

This is a variation of another question I posted. The difference is that here we have an equation that mostly works. We just need help finding how to push a Case. Honestly, we're just trying out ...
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1answer
34 views

characteristic function determines the distribution

I noticed that some probability and stats books talk about the characteristic function of a probability distribution of $\mathbb{R}^n$ and make the claim that the characteristic function uniquely ...
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1answer
33 views

Does centering the dependent variable and every independent variable change the estimated regressor?

In the linear regression model $Y=X\beta+u$ with $X=(\mathbf1, X_1, ...,X_p) \in\mathbb{R}^{n\times (p+1)}$ define $Y^*=Y-\bar{Y}$ and $X^*=(\mathbf0, X_1-\bar{X_1},...,X_p-\bar{X_p})$ Then \begin{...
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1answer
33 views

Counting probability question

I have a pretty basic statistics problem on multinominal distribution. Any feedback would be appreciated. If I have n marbles that can fall into three buckets (a, b, c) each with probability of $\...
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1answer
18 views

Is a random Gaussian distributed vector rotational invariant?

Given $v$ a random Gaussian distributed vector, is it possible to prove that $r = Rv$ is also Gaussian distributed? Any formula-link-information on how to start working on it is highly appreciated. ...
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3answers
86 views

Variance of dot product of two normalized random vector

Given a set of normalized vectors $\mathbf{x}$ and $\mathbf{y}$ of length $N$, with each entry independently sampled from $\mathcal{N}(0,1)$ before being divided by the vector norm. By running ...
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Study on the learning of programming concepts in pre-university students [closed]

I want to investigate the ability of a group of students to learn programming concepts in one hour. The students are grouped by ages: 6 - 17. The tasks are divided into concept and type. Each concept ...
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3answers
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In a survey of 100 people (population=1000), 100 people responded A and 0 people responded B. How to determine margin of error?

I took a few stats classes in university but it's been a decade since I did any of this so I apologize if this is trivial. I am looking at a research paper and 100 people were surveyed (total ...
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Probability of repeated events with fixed delay times co-occuring

I am trying to estimate the probability of signal collisions for a research project I am working on. The project uses acoustic emitters, with each emitter encoding a unique identification signal to ...
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1answer
31 views

Distribution of random variable Z=X/Y and conditioning

X is a continuous rv with pdf $f_X$ and cdf $F_X$ Y is a continuous rv with pdf $f_Y$ and cdf $F_Y$ Lets suppose X and Y independent and both positive. How do I go about finding the distribution of ...
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Produce an perpendicular vectors that follows a specific distribution

I am interested to produce a random Gaussian distributed vector perpendicular to a single or a collection of orthogonal predefined vectors. In the simple case of a single predefined vector, my first ...
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1answer
18 views

matching hats problem, expected value of number of matching hats

I've been reading a book where the following problem is stated "Suppose that $n$ people throw their hats in a box and then each picks up one hat at random. What is the expected value of X, the number ...
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2answers
50 views

Practical physical processes for various random distributions?

You can draw from the Cauchy distribution by attaching a stick to a spindle somewhere on the y-axis, spinning it, and reading off the x-intercept as your drawn value. Where you place the spindle on ...
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Expressing the following regression in terms of matrices depending on i

I am not sure, if this question belongs more to Cross-Validated or here. It's a mathematical question of a statistical regression. My aim is to express the following regression in terms of matrix ...
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3answers
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Intuition in the relation between force of mortality and survival function

I understand that The force of mortality $ \mu (x)$ can be interpreted as the conditional density of failure at age x given survival to age x, while f(x) is the unconditional density of failure at age ...