Questions tagged [self-study]

A routine exercise designed to test one's knowledge; often 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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Show there's no linear relation between X and Y

$\newcommand{\Cov}{\mathrm{Cov}}$ $\newcommand{\E}{\mathrm{E}}$ Show that if the joint of $X$ and $Y$, $f(x, y) = \frac{1}{2}\sin(x + y)$ for $0 \leq x, y \leq \frac{\pi}{2}$ and $0$ everywhere else, ...
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1 answer
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Shrinkage methods: does non-convex constraints ($q < 1$) also induce sparsity?

I'm reading ESLII, in particular the chapter about shrinkage methods, lasso, and ridge regression. The optimal model parameters for a given constraint $\sum | \beta_j |^q < \alpha $ are given by ...
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9 votes
3 answers
718 views

Studies with small sample sizes

I'm asking myself the question of why studies with small sample sizes are not as convincing as those with larger sample sizes, and when this becomes a statistical issue. A complaint I've heard a lot ...
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-1 votes
0 answers
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Your colleagues in a comp-bio lab have sequenced DNA from a large population in order to understand how a gene [closed]

Your colleagues in a comp-bio lab have sequenced DNA from a large population in order to understand how a gene (G) influences two particular traits (T1 and T2). They find that P(G) = 0.6, P(T1|G) = 0....
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1 vote
1 answer
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Covariance mixed model

Hi I'm learning about mixed model (2x2 cross over trial) and I don't understand how this works; How do we get to the last part of Covariance where Cov(Ek(i), Ek(i)) = Var(Ek(i)) = between subject ...
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0 votes
1 answer
38 views

Show an asymptotic property of a realization of an AR(1) [closed]

Let $$X_t = 0.9 \, X_{t-1} + \eta_t, \quad \eta_t \,\, \hbox{i.i.d.} \sim N(0,1)$$ Indeed, I want to show that a fixed realization $(\bar{X}_t)$ satisfies \begin{equation}\tag{I} \lim_{n \to \...
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UMP Test and UMVUE when there are nuisance parameters

Consider $X_1,...,X_n \sim Weibull(\theta, c)$ where $c>0$ is unknown. Several textbook examples consider when $c$ is known, but here, we consider when $c$ is unknown. Suppose now we wanted to find ...
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1 vote
1 answer
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Is there a formula for finding the number of ways 3 dice throws can add to 6?

Consider 3 dice throws: t1,t2,t3 Let E be the event that t1+t2+t3 add up to 6 What is the probability of event E? I know the denominator is an example of ordered repetition. So with n=6 and k=3 there ...
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1 vote
1 answer
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Showing that double integral of indicator random variable is the product of the random variables

Let $X$ and $Y$ be non-negative random variables with an arbitrary joint probability distribution function. Let $I(x, y) = 1$ if $X > x, Y > y$ and $0$ otherwise. Show that $\int_0^\infty \int_0^...
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0 answers
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What is the probability that the first number chosen is greater than the sum of the next three numbers chosen? [duplicate]

Let $ E_i\sim \mathcal N(20,3), i \in \{1,2,3,4\}.$ What is the chance that $E_1$ is greater than $E_2+E_3+E_4? $ I started by $\mathbb P(E_1>(E_2+E_3+E_4))=\mathbb P(E_1-E_2-E_3-E_4>0).$ What ...
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Why does the second term of predictive distribution variance vanish as $N$ approaches $\infty$?

The question comes from a statement in section 3.3.2 of Christopher M. Bishop's "Pattern Recognition and Machine Learning", 2006 edition. The related text is excerpted as follows: My ...
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1 vote
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Hypothesis testing misconceptions

I am reading a paper and it points out some common statistical misconceptions (the following are supposedly false): The p-value is the probability that you are taking the wrong decision if you reject ...
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0 answers
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LASSO with ill-conditioned sensing matrix

It is known that sensing matrix which meets the RIP criteria is a sufficient but not necessary condition for a guaranteed LASSO solution. I'm considering if a sensing matrix is ill-conditioned (with ...
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1 vote
1 answer
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What is the probability of two people who hit with probabilities p and q neither hitting their targets?

What is the probability of two people who hit with probabilities p and q respectively neither hitting their targets? A is the event that person one hits ~ Ber(p) B is the event that person two hits ~ ...
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-1 votes
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Neyman Pearson Lemma on Gamma distributions [closed]

There are two classes W1 and W2 and I have two different gamma distributions with known parameters: $p(x\mid w_1)=\frac{\alpha_1^{\beta_1}}{\Gamma(\alpha_1)}x^{\alpha_1-1}e^{-\beta_1x}$ $p(x\mid w_2)=\...
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2 votes
2 answers
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Understanding Equation (3.13) from Bishop's Pattern Recognition and Machine Learning

I'm having trouble deriving the mentioned equation from the text. Specifically, I'm having trouble obtaining the $\mathbf{\phi}(x_n)^T$ term. I checked the errata and while the lack of the $\beta$ ...
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1 answer
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Deriving the normal equations' coefficients

Suppose we use the least squares criterion to fit a linear model for the following dataset: $(x_1,y_1),...,(x_m,y_m)\in R \times R$, by solving the following optimisation problem: $$(a^*,b^*) = \text{...
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0 votes
1 answer
31 views

How can we incorporate an offset into a software library which performs least squares without offset

Disclaimer: this is not homework, just preparation for an exam I'll be taking How do I incorporate an offset term into a software library that performs least squares (without offset)? Let me give a ...
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0 votes
1 answer
26 views

What is the relationship between the derivative of a map and its image density? [duplicate]

Preliminaries Suppose we have a random variable $X$ with density $f$ and a suitably smooth function $g: \mathbb{R} \mapsto \mathbb{R}$. The random variable $Y = g(X)$ also has a density function $h$. ...
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Which of the following is an independent event? , or are there different types of independent events?

This is a high school math question: There are 6 balls in a bag, and one of them is red, and it is marked #1, 2 yellow ones are marked #1 & #2, 3 blue ones are marked #1, #2, #3 respectively。...
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Efron Hastie CASI exercise 4.3

This is an exercise from Computer Age Statistical Inference by Efron and Hastie, student edition of the book. Draw a schematic graph of $\dot{l}_{x}(\theta)$ versus $\theta$. Use it to justify (4.25) ...
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2 votes
2 answers
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Conditional expectation of the square of a random variable

Let the joint PDF of X and Y, $f(x, y) = \frac{1}{2}e^{-x}$ if $x \geq 0$, $|y| < x$, $f(x, y) = 0$ everywhere else. Calculate $\mathbb{V}(Y|X = x)$. By definition, $\mathbb{V}(Y|X = x) = \mathbb{E}...
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2 votes
2 answers
40 views

Regularisation Penalties/Norms

I've been working with some penalty-based models. Most specifically, LASSO. But I've also been reading surrounding literature on ridge and elastic net. Conceptually, I understand the nature of a ...
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1 vote
0 answers
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Solutions to 'Statistical Learning with Sparsity'

I've recently been working through Statistical Learning with Sparsity (SLS) by Hastie, Tibshirani and Hastie. I found some exercises very hard, and think I found some mistakes. A set of solutions ...
1 vote
1 answer
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I'm stuck in a part of an exercise where I'm asked to find a variance so that its confidence interval is within some bounds

I'm trying to complete an exercise from my book, asking me to find the variance to allow a sample of 49 items (that follows a normal law) to stay in bounds of one confidence interval, at 95% of ...
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-2 votes
1 answer
98 views

Mahalanobis $ d^2 $ is non decreasing

I have been studying multivariate analysis and hypothesis testing , and came across this question in back exercise Define Mahalabonis measure of distance squared between two populations with a common ...
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1 vote
0 answers
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Exponential Random Graph Model Implementation Python, Error [closed]

So I am trying to implement the most basic Exponential Random Graph Model implementation using Python. Concretely I'm trying to estimate the maximum pseudo likelihood estimate, which can be formulated ...
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0 votes
0 answers
20 views

Critical value for quality test

Consider a production company which produces tubes. The company manufactured of 13 batches of tubes. These batches can contain $4 \times 2200$, $5 \times 1700$ or $3 \times2500$ tubes per palette. ...
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3 votes
1 answer
34 views

Resources for learning about lasso

I'm looking for resources on lasso regression at the undergraduate mathematics level. All I can find is a lot of complex texts on variable selection, concentration inequalities, etc. I would like to ...
1 vote
1 answer
32 views

Internal consistency of correlation [duplicate]

I have this question below , and I am unable to understand what is internal consistency, can anyone please tell the concept , I have read its wiki page but I couldn't understand how to solve a ...
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2 votes
1 answer
179 views

Weight of a bag of crisps follows a normal law. Should calculating the probability that one weighs < 140 g take care if bags are in samples? [closed]

I'm stuck at an exercise question. A bag of crisps has a weight that follows $X \sim N(m=150\ \rm g, \sigma=20\ \rm g)$ we are taking samples of 15 bags of crisps whose have for mean weight $\overline{...
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6 votes
2 answers
244 views

Question on Linear Regression: What is a good transformation for this data?

$\newcommand{\rv}[1]{\texttt{#1}}$The Question: The wealth in billions of dollars for $232$ billionaires is given in $\rv{fortune}.$ Determine a transformation on the response to facilitate linear ...
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3 votes
1 answer
64 views

What test to use after a PCA?

Need some help answering a prac exam question "An ecologist is interested in the morphological adaptations of goannas to their microhabitat. She wishes to determine if goanna shape differs for ...
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0 votes
0 answers
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Sum of iid Exponential observations then subtracting the minimum of the observations [duplicate]

Consider $n$ iid $X_1,...,X_n \sim Exp(1)$. My goal is to find the density of $\sum (X_i - X_{(1)})$. My attempt If we write out the entire summation in order statistics, we get $X_{(1)}-X_{(1)} + X_{(...
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1 vote
0 answers
32 views

Understanding an outlier detection technique for fraud detection

I came across this article: http://projetoaprendizagemgrupo4.pbworks.com/f/03.03%20-%20Unsupervised%20Profiling%20Methods%20Fraud%20Detection.pdf since I am interested in detecting abnormal behavior (...
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0 votes
1 answer
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Why is the coefficient of determination less than or equal to 1?

I have been reading about the Coefficient of Determination and am wondering why it is necessarily less than or equal to 1. I understand that RSS is the sum of the difference between each dependent ...
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2 votes
0 answers
154 views

An asymptotic statistics calculation

I have a question about an apparently routine asymptotic calculation. Suppose that we have variables $X \in \mathbb{R}^{p}, Z \in \mathbb{R}^{q}, Y \in \mathbb{R}$. Write $E[Y \mid X] = f(X; \beta_0)$ ...
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0 votes
1 answer
31 views

How many ways can you arrange the digits: {1,2,2}, so that the resulting number would be even?

To quote the whole question accurately: Various 3 digit numbers can be formed by permitting the digits 122. All arrangements are equally likely. Given that any resulting number is even, in how many ...
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1 vote
1 answer
20 views

how to view conditioning on a random variable rather than a particular value of that variable

The question is about $P(Y|X)$ versus $P(Y|X=x)$. Is there an alternate way to write $P(Y|X)$ that makes its meaning more clear? I believe these are correct equations for conditional entropy: $$ H(Y|...
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0 votes
0 answers
21 views

How to build a Poisson random measure according a Levy Process

Let $(\Omega, \mathcal{F}, P)$ and $(\Theta, \mathcal{B}, \rho)$. A Poisson random measure (PRM) with intensity $\rho$ is a kernel $\mathcal{N}: \Omega \times \mathcal{B} \to \mathbb{R}$ such that: $\...
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5 votes
1 answer
84 views

Can there be functions $g$ and $f$ such that $\rho_{f(X),g(X+Y)}^2 > \rho_{X,X+Y}^2$

Motivation a special case A special case of this question is an inequality between the Spearman's rank correlation and the Pearson correlation (Why is the sum of individual Spearman's rho squared ...
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1 vote
1 answer
46 views

Is my application of Bayes' Theorem here correct?

I am new to Bayes' theorem and like many others, am having a hard time wrapping my head around it. Here's a question that I am trying to answer: Question: Imagine there's a bot detection tool that ...
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1 vote
1 answer
42 views

How to find the distribution of a function of two random variables?

I have the joint pdf $$f(x_1,x_2)=\begin{cases}\frac{1}{4}\;,\ -1<x_1,x_2<1\\ 0\;,\ \text{otherwise.}\end{cases}$$ I have to find the distribution of $Y=\frac{X_2}{1+X_1^2}.$ I have tried to ...
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2 votes
1 answer
90 views

Looking for a predictive prior for what I thought to be Bayesian Analysis

I need a prior with 2 parameters location (μ) and scale (σ) (a transformation of mean and standard deviation). My distribution is over discrete values, equally spaced with distance $c$ between ...
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3 votes
0 answers
33 views

Exact/Approximate Confidence Interval of Parameter Ratio from two Samples of iid Exponential

Suppose 2 independent samples $X_1,...,X_n \sim Exp(\lambda_1)$ and $Y_1,...,Y_m \sim Exp(\lambda_2)$, and are iid within samples. I am thinking about how to make an exact confidence interval for $\...
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0 votes
0 answers
19 views

Procedures to show that a process is not ergodic

I'm trying to show that a certain process is not ergodic, but as I don't have much experience, I would first like to learn how to show simple cases. We know that if a discrete stochastic process is i....
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1 vote
2 answers
75 views

Cross-validation by hand in R

Trying to teach myself cross-validation on a super simple example, linear regression. My understanding is that when I build a model via CV, it should have a lower RMSE. But I find almost identical ...
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  • 482
2 votes
1 answer
43 views

Covariance of the empirical probability mass function

Suppose a discrete random variable $Y$ takes $k$ levels of different values $y_1,y_2,...,y_k$. Let $P(Y=y_k):=p_k$. Suppose we have $n$ i.i.d. samples of $Y$, my question is: How can we compute the ...
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3 votes
1 answer
39 views

Boolean order statistic

Inspired from this LeetCode question. I authored this question myself however. Please review my answer for accuracy. Let's sort an array ($n \ge 1$) of $1$s and $0$s, so that all $0$s come before $1$s....
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1 vote
1 answer
51 views

Binomial Bayesian Regression with Metropolis Hastings in R

I'm trying to implement the Metropolis Hastings algorithm in this problem but I'm having problems with the convergence. $$Y_i|\beta_0,\beta_1 \sim \text{Binomial}(m_i,\theta_i)$$ where $logit(\theta) =...
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