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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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Simulate cycles [closed]

A single dump truck transports coal from a mine to an unloading facility near a railroad. Following the loading process at the mine, the truck promptly proceeds to a scale for weighing, and the ...
Deepika Walia's user avatar
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0 answers
23 views

Is there an additive property for hypergeometric random variables?

I know that for binomial and negative binomial RVs there is an additive property where if $X_1\sim bin(a, p)$ and $X_2\sim bin(b, p)$ then $X_1+X2 \sim bin(a+b, p)$ if $Y_1\sim NB(c, p)$ and $Y_1\sim ...
BadAtMaff's user avatar
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0 answers
28 views

Exam P Sample Problems # 30

The question: A company establishes a fund of 120 from which it wants to pay an amount, C, to any of its 20 employees who achieve a high performance level during the coming year. Each employee has a ...
BadAtMaff's user avatar
2 votes
3 answers
149 views

Conditional probability, who's right?

i need help with a math problem. Me, and at least two teachers have all done it and gotten different results, so now I'm asking the wise people of the internet for help settling the debate. Here's the ...
Azrael's user avatar
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4 votes
1 answer
214 views

Exercise about Order statistics from uniform distribution

I'm trying to solve an exercise about order statistics. The exercise is the following: Let $U_{(1)}< \ldots <U_{(n)}$ be the order statistics from Uniform distribution U(0,1). Show that $(-\log[...
MLe's user avatar
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0 answers
38 views

Likelihood function for data with random censoring

The following is from Klein and Moeschberger, p. 76. Let $(T,\delta)$ be a tuple with $T = \min(X,C_r)$ and $\delta = 0$ if the lifetime X is censored and $\delta = 1$ if it is not; $C_r$ denotes the ...
Montresor's user avatar
0 votes
1 answer
61 views

How to solve this "easy" probability problem

I'm struggling on how can I solve the following problem. I tried with a lot of calculus without success, but i think that the way to solve is: $$P(\text{at least one failure})=1−P(\text{no failures in ...
Ga13's user avatar
  • 280
1 vote
1 answer
84 views

Understanding the Logistic regression formula

Logistic regression aims at transforming the linear regression formula and fitting the s curve or logistic function to a particular dataset in order to calculate the probability of a categorical ...
Amelia Nicodemus's user avatar
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0 answers
35 views

Sufficient Statistic for Truncated Normal

I am doing exercise 3.18 of "The Bayesian Choice": Give a sufficient statistic associated with a sample $x_1,...,x_n$ from a truncated normal distribution $$ f (x|\theta) \propto \exp(-(x ...
daniel's user avatar
  • 143
0 votes
0 answers
22 views

Computing Bayesian model averaged posteriors

The Bayesian model averaged posterior predictive distribution for new data $\tilde{y}$ given training data $y$, across a set of $M$ models $\mathcal{D} = \{D_{1}, ..., D_{M}\}$, is defined as: \begin{...
user_15's user avatar
  • 185
2 votes
1 answer
40 views

Bayes Theorem and Total Probability

I'm back for more help....I'm terrible with conditional probability. Here's a problem from an assignment in a stats course I'm taking: A U.S. Army Corps of Engineers (USACE) study focused on DDT ...
jerH's user avatar
  • 235
1 vote
1 answer
39 views

Unsure about assumptions of linear model with time series variables, spurious regression and periodic patterns

Background I'm learning about time series in context of linear regression. The goal of this question is to understand how seasonality of either X or Y can affect the model. Linear model assumptions $...
Brzoskwinia's user avatar
1 vote
0 answers
46 views

What is the distribution of the unbiased estimator of variance for normally distributed variables?

I must be making some mistake in my derivation of the distribution of the unbiased variance estimator for i.i.d. $X_{i} \sim \mathcal{N}\left(\mu, \sigma^{2}\right)$. We have $\bar{X} =\frac{1}{n}\sum\...
YEp d's user avatar
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1 vote
0 answers
21 views

What is the formula to compute the height of significance line on autocorrelation plot?

Autocorrelation of a time series can be plotted in R with use of acf function. For example: acf(ldeaths) # built-in series I ...
Brzoskwinia's user avatar
0 votes
1 answer
27 views

Gaps (misinterpreting perhaps) between chi square sig and Cramer’s v

I did chi square test for independence and Cramer’s V test. I’m having problems with interpret the results. I checked if there is a connection between gender and dropping out of school (in SPSS). My ...
Shir's user avatar
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2 votes
1 answer
35 views

MLE of $\beta$ and $\sigma^2$ in linear regression with Laplace errors

Say we have a linear regression model $$ y_i = x_i^T\beta + \sigma\varepsilon_i $$ for $i = 1, \dots, n$, and $\varepsilon_i$ is distributed according to a Laplace distribution $\mathcal{L}(0, 1)$ ...
timeinbaku's user avatar
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0 answers
35 views

Complete second-order model with 2 quantitative and 1 qualitative variables

I want to check if what I have for the equation for the second-order model relating y to 2 quantitative and 1 qualitative variables (3 levels) is correct: $\mathbb{E}(y)= \beta_0 + \beta_1x_1 + \...
DragonFruit's user avatar
0 votes
0 answers
54 views

Usage of Sufficient statistic for a Gamma distribution

I need some help to understand how to utilize sufficient statistic from a data. Suppose I observe some random process that produces $x\in X$, where all elements have a gamma distribution. As far as I ...
tessob's user avatar
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0 votes
0 answers
51 views

Find $E[Y]$ when $f(x,y) = \frac{x}{3}e^{-xy}$

Truth be told, I don't really have an issue with this problem in general, but in it's calculation. Let me explain. We need to find $E[Y]$ when $f(x,y) = \frac{x}{3}e^{-xy}$, $1<x<4$ and $y>0$...
Anweshan Goswami's user avatar
3 votes
2 answers
179 views

Long-run average cost for uniform distribution

The lifetime of a device is a continuous random variable having the continuous uniform distribution $\mathrm{Unif}(0,15)$. Suppose that under an age replacement strategy a planned replacement at age $...
waterr's user avatar
  • 41
0 votes
0 answers
21 views

Lasso Regression Problem [duplicate]

$\operatorname*{argmin}_\beta\{\|y-X\beta\|^2 + \lambda\|\beta\|_1$, where $X$ is orthonormal. $\beta \in \mathbb R^d$. $X = [x_1,\ldots,x_n]^T$ and $y=(y_1,\ldots,y_n)^T \in \mathbb R^n$. $X^TX=I_{d\...
Harry Lofi's user avatar
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0 answers
67 views

Posterior distribution of shape & rate parameter in Poisson-Gamma Mixture

Currently I'm struggling to handle the following question. Suppose $x_i,(i=1,2,\dots,n)$ follows Poisson distribution: $$p(x_i|\theta) = \frac{\theta^{x_i}e^{-\theta}}{x_i!}, \quad x_i\in\mathbb N,\...
jason 1's user avatar
  • 311
0 votes
1 answer
38 views

question about outliers in a 5 term data set

the question states that no data set with only 5 terms has an outlier, and I'm stumped you're given a set of 5 numbers in ascending order $$ x_1, x_2, x_3, x_4, x_5 $$ I started by finding the IQR, ...
Bongo 186's user avatar
4 votes
2 answers
259 views

Expected value of decreasing function of random variable versus expected value of random variable

Given two random variables $X_1$ and $X_2$ (same sample space $\mathcal{X}$) that $$\mathbb{E}[X_1]=\int_{\mathcal{X}}xf_1(x)dx > \mathbb{E}[X_2]=\int_{\mathcal{X}}x f_2(x)dx$$ Can we say that $\...
Rokai's user avatar
  • 41
2 votes
2 answers
82 views

How to find probability from $E[X^n]$?

It is given that $E[X^n] = \frac{2}{5}(-1)^n + \frac{2^{n+1}}{5}+\frac{1}{5}$, where $n=1,2,3,\ldots.$ I need to find $P(|X-\frac{1}{2}| > 1)$. What my approach is : I have opened the modulus ...
Anweshan Goswami's user avatar
1 vote
2 answers
39 views

Interpretation of incidence rate ratio

The incidence rate ratio (and 95% confidence interval) of rotavirus gastroenteritis for the vaccine group compared to the placebo group is $0.67 (0.55, 0.82)$. I want to know approximately how many ...
Happy Cretine's user avatar
3 votes
0 answers
72 views

Posterior Distribution using Jeffreys prior

I'm trying to show that if $X_1, \cdots, X_n \stackrel{iid}{\sim} N(\mu, \sigma^2)$ with unknown $\mu$, $\sigma$ and the prior $\pi(\mu, \sigma^2) \propto 1/\sigma^2$ then the posterior distribution ...
S10000's user avatar
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0 votes
1 answer
58 views

Hastie "statistical learning" 2.28. Least squares and covariance

In Hasties book "statistical learning", just above equation 2.28, it says that $\mathbf{X}^T\mathbf{X} \rightarrow NCov(X)$ (when $N$ is large and $E(X)=0$). Why is this true? $Cov(X)$ is ...
LianLi's user avatar
  • 1
0 votes
0 answers
25 views

PDF of difference of uniform distributions [duplicate]

Main questions are in bold but feel free to correct me if I'm wrong somewhere else. As far as possible, I need both intuition and formal explanation. Let $X \sim Uniform(a,b)$ and $Y \sim Uniform(c,d)$...
White1Hun's user avatar
1 vote
1 answer
35 views

FInding a complete and sufficient statistic

I am attempting to learn how to find a complete and sufficient statistic. So, I am working on this problem for class: Let $X_1, \cdot\cdot\cdot,X_n$ be a random sample from the pdf $f(x_i|u)=e^{-(x-\...
Harry Lofi's user avatar
1 vote
0 answers
38 views

Understanding Schoenfeld Residuals

I'm reading the original paper by Schoenfeld from 1982. It is very laconic. Few parts that I don't understand: Why can we substitute $\hat \beta - \beta$ with the Score, and why is the score equal to ...
Maverick Meerkat's user avatar
0 votes
0 answers
33 views

GLM link function, variance Function, and dispersion for linear regression

we were recently introduced in class to GLM's and I am still trying to wrap my head around link functions, variance functions and dispersion paramters. In particular we were asked re-write the linear ...
Harry Lofi's user avatar
2 votes
0 answers
48 views

What to do in Box-Jenkins framework when time series has deterministic trend and seasonality?

I'm self-studying time series and I'm puzzled by apparent lack of consistency between : the "classical" decomposition of time-series and the Box-Jenkins methodology. Concerning the ...
Johannes Konrad's user avatar
0 votes
0 answers
40 views

BIvariate Normal and Conditional Expectation

I am working on a problem where I must show that the conditional distribution of Y given X follows the distribution with mean and variance shown below. In the previous question, we were given that X ...
Harry Lofi's user avatar
1 vote
1 answer
39 views

F-distibution only depends on degrees of freedom

I am reading ISLP Chapter 3.2 on multiple linear regressions. They define the F-statistic as $S_1$, the variance of sample 1 from population 1, divided by $S_2$, the variance of sample 2 from ...
amineh's user avatar
  • 87
0 votes
0 answers
52 views

Two-proportion hypothesis test with variances unknown

"A group of test subjects are split into two groups. 1000 people are given a vaccine, and 15 of them get the disease. 800 people are given a placebo, and 60 of them get the disease. Test the ...
Forklift17's user avatar
3 votes
1 answer
68 views

How to show in R, using a simulation, that when sampling from a normal distribution, the sample mean and sample variance are independent

What is the best way to show that when sampling from a normal distribution, the sample mean and sample variance are independent? I know the theory behind this result, I would like to show it using a ...
Carlos233's user avatar
0 votes
0 answers
51 views

F-Statistic and F-distribution [duplicate]

I am reading ISLP chapter 3.2 about multiple linear regressions. F-statistic seems like to be a statistic calculated per sample. But also we know that it is just and estimation for the stat on whole ...
amineh's user avatar
  • 87
1 vote
0 answers
41 views

Finding probability involving dependent random variables [closed]

Suppose train on line A arrives in time uniformly distributed between 0 and 4mins, train on line B arrives in time uniformly distributed between 0 and 6 mins, and furthermore the time interval between ...
Harsh's user avatar
  • 11
2 votes
1 answer
68 views

Exercise involving Bayes' Theorem

Consider the problem During exam time, in a certain school, only 25% of the teachers warn their students in writing that they are not allowed to get up to ask questions during exam time. in writing to ...
Wrloord's user avatar
  • 165
3 votes
2 answers
92 views

Find $P(Y>\frac{5}{4})$ when joint pdf of $X$ and $Y$ is given

The joint pdf of $X$ and $Y$ is given as $f(x,y)$ = $\frac{1}{2x^2y}$ , $1<x<∞$ and $\frac{1}{x}<y<x$ Then find $P(Y>\frac{5}{4})$ Since the limits of $Y$ have $X$ in them, I am not ...
Anweshan Goswami's user avatar
7 votes
3 answers
580 views

How do I compute a probability from the MGF?

I have a random variable $X$ with moment generating function: $$m_X(t) = \frac{2}{9} + \frac{e^{-t}}{9} + \frac{e^{-2t}}{9} + \frac{2e^{t}}{9} + \frac{e^{2t}}{3}.$$ I want to find the probability $\...
Anweshan Goswami's user avatar
4 votes
1 answer
145 views

Comparison of confidence intervals: bootstrap & exact resampling

Consider data $X_1,...X_n$ generated from a probability distribution $F$ with density $f$. I'm interested in constructing confidence intervals for a parameter say, $\theta(F)$. Via Monte Carlo ...
reyna's user avatar
  • 365
3 votes
1 answer
149 views

If X ~ Bin(2, 1/2) and Y ~ Bin(3,1/3), then find P(2X+3Y=13)

The 2 random variables have different parameters p, therefore cannot be added. I understand that 2X ~ Bin(4,1/2) and 3Y ~ Bin(9,1/3) [by additive property of binomial distribution with same parameter ...
Anweshan Goswami's user avatar
0 votes
1 answer
29 views

Principal Component Analysis and Relation to the SVD of a matrix [duplicate]

We are learning about Principal Component analysis in our class, and I having trouble understanding how to compute the principal component given a matrix. For example, here is the matrix we were given....
Harry Lofi's user avatar
7 votes
1 answer
108 views

Power of two-sample z-test

In a pilot study with two groups, the control distribution has the Mean1 = 90, SD1 = 5 and the treatment distribution has Mean2 = 85, SD2 = 5. The null hypothesis of the test is that the sample ...
Christian's user avatar
  • 193
0 votes
0 answers
24 views

Does this state space model make sense?

I'm working on a problem, Consider that a times series $\{y_t\}$ is generated from an $\text{ARIMA}(1,1,1)$ model, so that $$y_t-y_{t-1}=\alpha(y_{t-1}-y_{t-2})+\epsilon_t+\gamma\epsilon_{t-1},$$ ...
mjc's user avatar
  • 589
1 vote
1 answer
47 views

What modeling approach should i use AR, MA, ARMA?

those are my ACF and PACF plots for my time series after two differentiating. I watched a couple of tutorials but I cannot figure out what method I am supposed to use. Also, an interpretation of the ...
antekkalafior's user avatar
1 vote
0 answers
119 views

Distribution of $F_n^{-1}(3/4)-F_n^{-1}(1/4)$ [closed]

Given $X_1,X_2,...X_n\overset{\text{iid}}{\sim}F$, find the distribution of the sample inter quartile range, $F_n^{-1}(3/4)-F_n^{-1}(1/4)$ in terms of $F$ where, $F_n$ is the emperical distribution ...
reyna's user avatar
  • 365
3 votes
2 answers
196 views

Calculating $E[(\sum X_i)^4]$

Trying to figure where I'm going wrong with the following. My goal is to calculate var$(\bar X_n^2)$ using $E[(\bar X_n)^4]=\frac{1}{n^4}E[(\sum X_i)^4]$ given that $X_1,...X_n$ are iid with $EX_1=\mu,...
reyna's user avatar
  • 365

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