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Questions tagged [self-study]

A routine exercise from a textbook, course, or test used for a class or self-study. This community's policy is to "provide helpful hints" for such questions rather than complete answers.

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OLS basic doubt

In a multivariate OLS model : $ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \epsilon$ My estimator for $\beta_1$ is given by which expression: $\hat \beta_1 = [X_1'X_1]^{-1} X_1'Y$ OR $\hat \...
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21 views

Expectation of a product of random variables problem

Let $U$ be a random variable with uniform distribution over $[0,1]$ and a bivariate random vector $(Z,T)$ defined by $$(Z,T)=(0,0)1_{\{U\geq 2\alpha\}}+(Q_X(U),Q_Y(U))1_{\{U< 2\alpha\}}$$ where $...
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1answer
19 views

How much matrix differentiation background for Seber and Lee's Linear Regression Analysis

How self-contained is Seber & Lee's textbook 'Linear Regression Analysis'? It seems to assume knowledge of matrix differentiation.
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46 views

Stochastic Sequence; Compute $\lim_{n\to\infty}$

Let $(X_n)n≥1$ be a stochastic sequence of i.i.d. random variables, each $X_n$ with values in the set {1, 4, 8, 16} and probability distribution: $P[X_n = 1] = 1/6, P[X_n = 4] = 1/4, P[X_n = 8] = 1/3,...
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36 views

Multiple linear regression: am I interpreting the methodology right?

This is a follow-up question to 1 and 2. So we have the normal linear model \begin{align*} \textbf{Y} = \textbf{X}\beta + \epsilon \end{align*} where $\epsilon\sim\mathcal{N}(\textbf{0},\sigma^{2}\...
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16 views

law of iterated expectations doubt

Is it correct to do so: $E(X^{-1} Y) = E_x(E(X^{-1} Y \vert X) = E_x(X^{-1} E(Y \vert X)) $ Are we allowed to use LIE in case of non linear functions like $X^{-1} $
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13 views

i.i.d. random vectors [duplicate]

If $(X_1,Y_1), (X_2,Y_2)$ are independent random vectors having the same joint distribution function $F$, then is it correct to say: $E(X_1)=E(X_2)$ and $E(Y_1)=E(Y_2)$ (the same for variance); Both, ...
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1answer
37 views

regression basic doubt

If have a simple bivariate regression model: $ Y_i= x_i \beta + \epsilon_i $ where $i$ are the number of observations. How do I test for the hypothesis that the OLS coefficient $\beta$ does not ...
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1answer
50 views

Why do we need to determine the distribution of $\textbf{Y}$ in the multiple linear regression problem?

Once again, here I am. Given the multiple linear regression model \begin{align*} \textbf{Y} = \textbf{X}\beta + \epsilon \end{align*} where $\epsilon\sim\mathcal{N}(\textbf{0},\sigma^{2}\textbf{I})$ ...
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1answer
24 views

Testing whether two categorical variables have identical coefficients

I am currently doing an exercise question asking me to construct a model to test whether the coefficients of two categorical variables ($X_2$ and $X_3$) are same in R. Specifically, these two ...
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1answer
30 views

Question about the Multiple Linear Regression: why and how does it work?

I know this question is quite simple and maybe quite naive as well, but I would like to get some help. The general linear model can be expressed as \begin{align*} \textbf{Y} = \textbf{X}\beta + \...
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1answer
48 views

How to choose the best method to generate random values [on hold]

In my specific case, I have a pdf that has no closed form, and I want to generate random values ​​of this distribution. It depends on a summation that goes to infinity (coming from a poisson process) ...
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14 views

Gradient clipping just before averaging

A typical way of implementing mini batch learning is by calculating the gradients of every element within the mini batch and then average all of these element's gradients to come up with the final ...
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34 views

No population? How to answer such question?

Say the givens are: probability of a battery being defective is = 0.40 being not defective = 0.60 STD = 1.5 hours mean = 7.5 hours The question states: if 5 batteries are randomly selected what is ...
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26 views

Is assigning an inverse-Wishart distribution to a diagonal matrix problematic?

I'm reading the paper Bayesian Vector Autoregressions by Thomas Wozniak. He considers the model $$y_t = \mu + A_1 y_{t-1} + \cdots A_k y_{t-k} + u_t$$ where each $y_i$ is a $N$-vector, each $A_j$ is a ...
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6 views

Lagged Panel Data in R [on hold]

I want to regress a panel data using within estimator in R For the static model, I have yit = B1*x1,it + B2*x2,it + eit Now I want to estimate the dynamic model yit = B0*yit-1 + B1*x1,it + B2*x1,...
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1answer
35 views

Stationary Process Ergodicity

Can you give me an example of a stationary nonergodic stochastic process that is time continuous?
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1answer
34 views

pdf from a set of conditional pdfs

I have an interesting problem, i have seen in many text books ways of calculating conditional pdfs but not many where given a set of conditional pdfs for a variable we wish to calculate it's pdf. In ...
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0answers
7 views

Rank aggregation [on hold]

I am working on an independent project that requires me to aggregate rank. There are multiple categories and people are assigned ranks(based on raw scores) within each category and the individuals are ...
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0answers
14 views

Condensing values of categorical data

Beginner ML question here. I have a dataframe with a categorical column, a lot of the values are slightly different but essentially mean the same thing. Here's an example of such values: ...
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41 views

How can I fit a multiple linear regression model in R if the value for beta coefficient of each predictor is given? [on hold]

I've got an exercise question asking me to fit a multiple linear regression model in R when the values of coefficients are given. I don't know how to do it. specifically, I have 5 predictors in my ...
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1answer
53 views

Computing Variances by Conditioning

I have trouble with the first part of this problem (Please take a look at the image below). This is an example problem from my old textbook years ago and I have had trouble understanding: How Y is ...
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42 views

How do you define the sensitivity of a neural network? [on hold]

What are the sensitivities of a neural network with a sigmoid output node, and two relu hidden nodes? In this context, by "sensitivity" I mean he sensitivity of the function with respect to that ...
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29 views

expected value (Sheldon Ross 9th edition)

A person buy 10 lottery tickets with each having a winning probability p. If he wins a prize in atleast one of the tickets,he gets addicted and will keep on buying tickets till he wins again. Find the ...
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9 views

1-NN Discriminant Function and Classification Error

The question is as follows: Consider a K-NN classifier where k=1. Assume we have two classes $C_1$ and $C_2$ with equal priors $P(C_1) = P(C_2) = 0.5$. We are given data from the two distributions: $...
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28 views

Exponential family and geometric distribution: how do we prove the sum of independent geometric random variables has negative binomial distribution?

Let $X$ be the number of failures before the first success in a sequence of Bernoulli trials with probability of success $\theta$. Then $P_{\theta}[X = k] = (1-\theta)^{k}\theta$, $k = 1,2,\ldots$ ...
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29 views

variance of multinomial distribution

Assume $A_{kj} \sim$Multinomial$(1, \;\underbrace{(1/m, 1/m, ..., 1/m)}_{\textrm{m times}})$, where $k=1,2, ... m$ and $j=1,2, ... n$. It is clear to see that $\sum_{k=1}^mA_{kj}=1$. If we impose a ...
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20 views

variance of restricted random vars

Let $X_1$ and $X_2$ are i.i.d. with Bernoulli($p$). If we are interested only on a set of realizations of $X_1$ and $X_2$ denoted by $Y_1$ and $Y_2$, respectively, such that $Y_1+Y_2=S$, then what ...
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1answer
12 views

Prerequisites for Latent Dirichlet Allocation

I have read several "intuitive" introductions to LDA. However, I now want to learn it properly. I have already read through most of Duda, and that was my introduction to data science. However, I ...
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30 views

Confidence interval for population variance and chi-square distribution [closed]

At present, i am reading the e-book by Robert L. Mcdonald(Professor of Finance,Northwestern University's Kellogg school of Management) named $"Derivatives Markets" 3rd Edition.$ In that book, I didn'...
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Why do people use PCA when it has so many issues?

(This is a soft question) Recently I'm learning Principal Component Analysis, and it appears to have a lot of issues: You have to transform the data to roughly the same scale before applying PCA, but ...
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1answer
862 views

If the square of a time series is stationary, is the original time series stationary?

I found a solution that stated that if the square of a time series is stationary, so is the original time series, and vice-versa. However I don't seem able to prove it, anyone has an idea if this is ...
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90 views
+50

Test for Lipschitz continuity (is there some?)

Let $x_1, \dots, x_n$ be a random sample from a distribution $D$. Say, I want to test whether $F(z)$, the cdf of $D$, is Lipschitz continuous, i.e. there exists $L$ such that $F(z + \delta) - F(z) \...
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Probability distribution question

An auto parts company, produces cylinder liners for engines of 1.2 inches in average diameter with a standard deviation of 0.1 inches. Every piece has a diameter less than an inch or more than 1.4 ...
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42 views

Finding the posterior distribution of a Bayesian analysis prior

I have a prior distribution $f(x)=\pi cos(\pi x) $ where $x$ is the probability of getting tails in a coin toss. Should a coin toss result in tails, how would this be reflected in the posterior ...
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34 views

Showing $Y_n\stackrel{p}\to Z$ where $Y_n=B_nZ+(1-B_n)X$

I am reviewing some of my old class notes again, and I came across the following problem. I think I have solved the problem correctly, but I wanted to see what others here thought. Do you think I ...
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26 views

Recurrence of $k$-step ahead forecast with ARMA

For brevity, let's consider an AR(1) model, but this question should apply to ARMA(p, q) in general. Assume we are at time $T$ and would like to forecast $k$ steps ahead, $$ X_{T+k} = \phi_0 + \phi_1 ...
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1answer
50 views

Two-sided UMP test for exponential densities?

I'm struggling with a problem from Lehmann & Romano's book *Testing Statistical Hypothesis." Suppose $X_i$ is a random sample from $$f(x) = \frac{1}{b}e^{-(x-a)/b}\mathbf{1}_{x>a}$$ The ...
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3 views

Soft margin SVM seperability

I have a set of data points of the following form: - - - + + + + if $C=0$ then how many support vectors do we have? if $C= \infty$ then how many support vectors do we have? if $C=0$ then its a ...
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2answers
103 views

MCMC algorithm going wrong [closed]

Given this integral \begin{equation} \int_0^\infty \chi_{[1,2]}(x)\Gamma(C,x)\left|\cos(R x)\right| \, dx \end{equation} where $\chi_{[1,2]}=\begin{cases}1, & x \in [1,2] \\ 0, & x\not \in [...
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8 views

Different classification loss for K-nearest neighbours

Suppose we have a general classification loss instead of a 0-1 loss. How can we modify k-NN to accommodate such a loss function? I thought about using a weighted loss matrix where $L(i,j)=0$ when $i=...
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1answer
61 views

Finding the characteristic function of a simple linear PDF

X is a random variable with a pdf of $ f(x) = \begin{cases} x/2, & 0 \le x \le 2 \\ 0, & \text{otherwise} \end{cases}$ I tried finding the characteristic function of this but ended up with ...
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How do I find the probability of interference of a shaft and a bearing given their nominal diameter values and their standard deviations?

The exact question is as follows: An assembling of shaft and bearing is made out of shaft manufactured to a specification of 30.00 ± 0.09 mm and bearings are manufactured to a specification of 30.10 ±...
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1answer
45 views

Why should we study copulas? [closed]

I am new to the study of copulas and I would like that someone could provide some examples where they are applied, their usefulness and so on. Any help is appreciated. Thanks in advance.
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27 views

What receptive field do we have after stacking $n \times n$ CONV layers with kernel size $k \times k$?

What receptive field do we have after stacking $n \times n$ convolutional layers with kernel size $k \times k$ and stride $1$? Layers numeration starts with $1$. The resulting receptive field will be ...
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1answer
82 views

Maximum Likelihood Estimator of $P(Y_1=1)$ where $Y_i=1$ if $X_i>0$ and $0$ otherwise, given $X_1,\dots,X_n\sim N(\theta,1)$

This is part(a) of exercise 6 of Chapter 9 from Wasserman's All of Statistics. Let $X_1,\dots,X_n\sim N(\theta,1)$. Define $Y_i=\begin{cases} 1 &\text{ if }X_i>0 \\ 0 &\text{ if }X_i\le 0....
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Time Series: ACF and PACF plot, how to tell what's the best model by looking at the plot?

The question ask me "The equation of the model you think is most appropriate, given the plots. Justify your choice of model. " the ACF plot looks like cut off after the first lag. And I'm not too sure ...
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1answer
37 views

Does independence and mutual exclusivity induce impossibility?

Given that we know A and B are independent and they never occur at the same time, one of them must be impossible, no? $$ P(A\mid B)=\frac{P(A \cap B)}{P(B)}\\ \text{if A and B independent, B gives no ...
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1answer
32 views

Unbiased variance question [closed]

A researcher is testing if a new swimming technique is more effective. She knows the average 50m time of swimmers in her club using the old technique is 35 seconds. After training 12 swimmers with the ...