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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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
2 votes
1 answer
44 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
44 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
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1 vote
0 answers
34 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
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1 vote
1 answer
55 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
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2 votes
2 answers
80 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
388 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
132 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
  • 353
2 votes
1 answer
58 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
0 answers
13 views

Statistics in application and for calculation as separate or together? [closed]

I need statistics and the application in R for my studies, which is at the level of social sciences. Personally, I have found using exercises in ...
user406868's user avatar
0 votes
1 answer
25 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
5 votes
1 answer
69 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
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0 answers
22 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
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1 vote
1 answer
38 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
118 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
  • 353
3 votes
2 answers
183 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
  • 353
1 vote
3 answers
80 views

Asymptotic normality implies consistency

I'm trying without success to solve the following exercise in my econometric textbook: Show that $\sqrt{N}\left(\widehat{\beta_1} - \beta_1 \right) \xrightarrow{d} \mathcal{N}(0,a^2)$, where $a^2$ is ...
Residual Claimant 's user avatar
1 vote
1 answer
26 views

How to manually evaluate a parameter estimate using the first few values of a time series?

I'm working on a problem (I paraphrase), Consider the $\text{AR}(2)$ model $$y_t=\alpha y_{t-1}-(1-\alpha)y_{t-2}+\epsilon_t, \hspace{1em} \epsilon_t\sim N(0,\sigma^2).$$ The conditional least ...
mjc's user avatar
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1 vote
0 answers
43 views

Deriving Sample Size for two samples t test

I am trying to figure out what sample size recommendations to make in this case when the samples have different sizes. Since their sizes are different the variance of each will be different. Suppose ...
Harry Lofi's user avatar
4 votes
2 answers
74 views

In R, why are the results of summary(lm(y~-1))$sigma^2 and mean(y^2) same?

Consider the following data and the model: y <- c(9,3,-2,7,6,12) lm(y~-1) summary(lm(y~-1))$sigma^2 [1] 53.83333 I expected ...
user232597's user avatar
3 votes
1 answer
164 views

Finding the MLE for a piecewise function

I am attempting to find the MLE for the two parameters (i.e. $\alpha, \beta$) in the following piecewise function, but I am slightly confused as to how to proceed through this question: $$ P(X_i \leq ...
Harry Lofi's user avatar
0 votes
0 answers
36 views

How to derive OLS estimates of beta by minimizing the error sum of squares?

For the model: $Y_i=\beta_0+\beta_1X_i+\epsilon_i, \epsilon_i \sim^{iid} N(0,\sigma^2)$ for $i=1,...,n.$ Let $X_i >0$ and $\sigma^2_i=\sigma^2X_i$. $\tilde{Q}(\beta)=\sum^{n}_{i=1}\sigma_{i}^{-2}\{...
Harry Lofi's user avatar
2 votes
1 answer
59 views

Simple vs. Composite Hypothesis Question

I am looking at one of the questions in the qualifying exam, where a multiple logistic regression model is being fitted, where the response is about dying. The question says that there are multiple ...
Texas2022's user avatar
1 vote
0 answers
63 views

How to deal with a indicator function in a likelihood? [duplicate]

Question: A panel study followed 25 married couples over five years. One item of interest is the relationship between divorce rates and the various characteristics of the couples. For example, the ...
Irohas's user avatar
  • 11
3 votes
1 answer
61 views

How to write the AR polynomial for a model containing a constant?

I'm working on an exercise, Consider the autoregressive time series model $$y_t=c+\frac{1}{3}y_{t-1}-\frac{1}{2}y_{t-2}+\epsilon_t,\tag{$\ast$}$$ where $\epsilon_t$ is white noise with variance $1$. ...
mjc's user avatar
  • 589
0 votes
0 answers
35 views

How to draw samples from two correlated Negative Binomial variables?

Data and problem description: my data is the number of corner kicks of home team, away team, total corner kicks and corner kicks difference. Below is code for data and the plots (assuming the number ...
Juan's user avatar
  • 57
0 votes
0 answers
19 views

Single or Multiple Bivariate analysis?

For the data structure, we are fitting the multilevel model to the data. Before fitting the model, we are eager to do bivariate analysis so that we can keep those independent variables in the ...
user232597's user avatar
1 vote
0 answers
68 views

Correct interpretation of Bayesian credible intervals

I'm starting to learn about Bayesian statistics for my own personal education (this is not a homework question). Something puzzles me a bit about the interpretation of credible intervals. From the few ...
Daniela's user avatar
  • 57
1 vote
1 answer
22 views

Getting a very big relative risk ratio value

We have fitted a multilevel multinomial logistic regression model to our data. We have obtained relative risk ratios(RRR). For most of the independent variables RRR have usual values, like 0.49, 0.78, ...
user232597's user avatar
3 votes
2 answers
324 views

Interpreting when a regression coefficient is significant

Consider the following regression model: $y_i=\beta_1+\beta_2x_{i,2}+\beta_3x_{i,3}+\beta_4x_{i,2}x_{i,3}+\epsilon_i,$ where $\epsilon_i\sim N(0,\sigma^2).$ Here, $x_2$ is binary variable $$X_2 = \...
user232597's user avatar
3 votes
4 answers
651 views

What is the difference between sampling error and measurement error?

The terms sampling error and measurement error in statistics field are very relevant and have a paramount importance for data analysis and reporting.
Dr. Subhash Chander's user avatar
4 votes
3 answers
304 views

Deriving the Distribution of Markov Chain Times

I am interested in learning how to derive the probability distributions for the Time to Absorption in Markov Chains (Discrete and Continuous). In the past, I have usually done one of the following: ...
Uk rain troll's user avatar
2 votes
1 answer
28 views

Identifying the alternate hypothesis [basic]

I was dealing with the following question: After a change management, a producer claimed that less than 5% of his flak jackets have production flaws. A sample of 200 flak jackets was randomly ...
Celine Harumi's user avatar
0 votes
0 answers
34 views

Derive LM test for linearity

I am studying nonlinear models. And I try to derive LM test for nonlinear model against well-specified parametric linear models The book gives score vector $s(\theta)$ and information matrix $I(\...
1190's user avatar
  • 1,060
1 vote
0 answers
108 views

Tightness of rejection sampling

Hello. I'm studying the Monte Carlo Statistical Method textbook by Robert and Casella. I have a question about exercise problem 30 in Chapter 2. I've already solved parts (a)-(c), but I'm having ...
Kim Gwang Woo's user avatar
1 vote
1 answer
57 views

Help with Gambler's ruin problem

Players A and B each have $10 at the beginning of a game in which each player bets at each play, and the game continues until one player is broke. Suppose that Player A wins on a single bet with ...
waterr's user avatar
  • 11
1 vote
0 answers
63 views

$U$-statistics and their limiting distributions

Let $X_1,X_2, . . . ,X_n$ be i.i.d. observations from a continuous distribution $F$. Consider the parametric function $\mathbb{P}([\text{min}(X_1,X_2) > X3])$. Find the U-Statistics and its ...
user771946's user avatar
0 votes
0 answers
10 views

What's the purpose by returning back the bad outliers from AdaBoost?

Assume that you having a matrix $X$ that holds both numerical and binary data. You plug in the data $X$ into AdaBoost and AdaBoost update the $X$ by only focusing on the rows that could not make a ...
euraad's user avatar
  • 415
1 vote
0 answers
49 views

Construct an estimator for the asymptotic variance of the MLE and and its asymptotic distribution

This is a question I have come across while learning about statistical inference and data analysis. I think I have been able to solve question (a) and (b) already. My solutions are: $$ \hat{p}_{MLE} = ...
Lucius Aelius Seianus's user avatar
-1 votes
1 answer
89 views

Hypothesis testing of categorical variable

I have extracted data from a questionnaire about factors influencing employability of young graduates. The questionnaire was answered by students in my class, so I extracted their age, major, and ...
HellBoy's user avatar
  • 123
2 votes
1 answer
47 views

Comparing statistical measurements of several countries

In a study, an outcome variable has three categories. There are several factors in the study. We are determining how these factors influence the outcome variable.We have fitted multiple logistic ...
user232597's user avatar
3 votes
2 answers
89 views

How to calculate probability of a team winning in penalty kick?

I was browsing the web and I saw this chart. I understand penalty rules but I really have no idea how to calculate the numbers in the chart, right on the first line. My idea is to consider each shot ...
Xero0808's user avatar
0 votes
0 answers
79 views

Finding joint probability distribution of standard normal with constraints

Let $X, Y \mathop{\sim}\limits^{iid} N(0,1)$. a) Suppose $X < Y$, find the joint pdf of $X$ and $Y$. b) What is the joint pdf of $X$ and $Y$ if $X = Y$? We know that $f_X(x) = \frac{1}{\sqrt{2\pi}}...
Hammed's user avatar
  • 1
1 vote
0 answers
45 views

UMP size $\alpha$ hypothesis test for discrete distribution

Let $\left(Z_1, \ldots Z_n\right)$ be an i.i.d. sample with common probability mass function $$ p_Z(j)=\vartheta^2 I_{[j=1]}+2 \vartheta(1-\vartheta) I_{[j=2]}+(1-\vartheta)^2 I_{[j=3]} $$ Find a UMP ...
Stats_Rock's user avatar
0 votes
0 answers
71 views

Interpretation of PCA principal components

As a result of a PCA analysis with some initial correlated variables $X_1,..,X_k$ we often do a plot of the loadings to intepret the principal components: PLOT 1: Plot of the loadings: We call $v^i,i=...
Thomas's user avatar
  • 870
3 votes
0 answers
17 views

Apriori Algorithm confusion: difference between frequency and support

What is the correct way to calculate support? I have seen two different ways and I'm confused as a result. One way (say first approach) is explained https://en.wikipedia.org/wiki/...
Srishti M's user avatar
  • 1,315
2 votes
1 answer
57 views

LMM in SPSS for a medical treatment analysis (repeated measurements and nested structure)

For a statistical analysis of a medical treatment I want to use an LMM (linear mixed model) and am quite unsure on how to do that correctly, in particular in SPSS (which is the license I have obtained ...
Sellerie's user avatar
  • 121
0 votes
0 answers
38 views

Hypothesis Testing of categorical variables [duplicate]

CONTEXT Suppose i collected data sets from a survey over google form, my sample is consisting of 32 row with the age/sex/major of the person who filled the survey, they a multiple choice answer for ...
HellBoy's user avatar
  • 123
1 vote
0 answers
20 views

During Neural Network training, training loss decreases then hits an inflection point and increases. What are some possible explanations?

"During Neural Network training, training loss decreases then hits an inflection point and increases. What are some possible explanations?" I got this question in an interview and wondered ...
Gooby's user avatar
  • 149
1 vote
0 answers
14 views

How to derive the Sum of Square for Latin Square with repetition

I am studying the Latin Square Design with n repetitions from https://online.stat.psu.edu/stat503/lesson/4/4.4 Let say we talk about the repetitions case 3 i.e. we ...
Brian Smith's user avatar

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