# 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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### 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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### 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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### 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
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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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### 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 \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
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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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### 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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### 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
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### 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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### 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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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....
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) =...