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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.

2
votes
2answers
31 views

UMVUE for probability of cutoff

Let $X_i \sim N(\mu,1)$, i.i.d. We aim to find UMVUE for $p(\mu) = P_{\mu}(X_1 \leq u)$ for some fixed $u$. I have shown that $\bar{X}$ and $X_1 - \bar{X}$ are independent. ($\bar{X}$: sample mean). ...
0
votes
0answers
14 views

Determining the limiting distribution of MLE

Let $X_i$ be i.i.d. with density $p(x) = ce^{-x^4/12}$, $c$ is a normalizing constant. Consider the location model $p(x-\theta)$. Aim is to compute the limiting distribution of the MLE. If I have ...
1
vote
1answer
21 views

ARIMA vs SARIMA

I am a self learner, and I am studying time series analysis. I came through the fact that ARIMA can be used to model a time series which is not stationary (Integrated ARMA model). The non ...
0
votes
0answers
34 views

Does the Markov property always hold for a state-space structure?

Markov Property: $p({\bf x}_t | {\bf x}_1, \ldots, {\bf x}_{t-1}) = p({\bf x}_t | {\bf x}_{t-1})$ Consider the following model for which the hidden states are ${\bf x}_t$ and the observations are ${\...
0
votes
1answer
30 views

Stating a confidence interval for odd ratio

A research institute surveyed 1000 online shoppers in the United States and China. One question asked if the online shopper followed brands they purchased through social media. Here are the results <...
0
votes
0answers
17 views

Which statistical methods to be used?

I am analyzing data in order to show the reasonings of interindividual variabilities of the results of a laboratory data that was measured repeatedly over 100 hours in a single group. But the timings ...
2
votes
1answer
40 views

How to express a correlation matrix in terms of a covariance matrix?

I'd like to know, if I have the following Normal multivariate structure $$\left( {\begin{array}{*{20}{c}} {{Y_{r_1}}}\\ {{Y_{r_2}}} \end{array}} \right)\sim{\mathcal{N}_{r}}\left( {\left[ {\begin{...
1
vote
2answers
46 views

Irreducible markov chain problem

Let $P$ be a stationary transition probability matrix of the markov chain $ \{X_n , n \ge 0\}, $ which is irreducible and every state has a period 2. Further suppose that the markov chain $\{Y_n , n \...
-1
votes
1answer
17 views

Probability proportional to size

Consider the problem of drawing a sample of size $2$ from a finite population of size $20$ . The sampling is done with replacement using probability proportional to size scheme . The normed size ...
0
votes
0answers
12 views

BIBD MCQ PROBLEM [on hold]

Consider a balanced incomplete block design BIBD with parameters $(b,k,v,r,\lambda)$ , where the symbols have their usual meaning ,which of the follwoing can not possibly be the parameters of BIBD ? ...
0
votes
0answers
12 views

RSS in Johnston-Dinardo

In Johnston-DiNardo it is stated that: But on the very next page it states: My professor applyed the second one for computing $$s^2=(e'e)/(n-k)$$ Where $$e'e=y'y+b'X'Xb$$ Why this distinction? ...
-2
votes
0answers
13 views

Prospective Cohort Study [duplicate]

In a prospective cohort study three hundred thirty-seven patients had HCV RNA(chronic infection), 91 had HCV antibodies and no HCV RNA(cleared infection), and 1957 had no HCV markers. Median CD4(+) T-...
-1
votes
0answers
16 views

Epidemiology-Prospective Cohort Study [on hold]

In a prospective cohort study three hundred thirty-seven patients had HCV RNA (chronic infection), 91 had HCV antibodies and no HCV RNA (cleared infection), and 1957 had no HCV markers. Median CD4(+)...
2
votes
1answer
57 views

How to use Gamma distributions to estimate the number of failures?

I need to calculate the expected number of failures of a product within 6 years. The time until failure is said to be gamma distributed with $\alpha=2$ and $\beta=0.5$ for a mean time between ...
0
votes
1answer
46 views

Conditional Mean E[X|X>Y] when X,Y are Uniform [duplicate]

X and Y are independent and identically distributed uniform (0,1) random variables. Find E(X|X>Y) Please explain it step by step. I am getting ans 0.5 while the actual answer given is 0.66
-1
votes
1answer
21 views

Exponential MGF

Let $T$ be a random variable with the probability density function (PDF) $f(t) = \lambda e^{−λ(t−a)}$, $\lambda > 0$, $t > a$. Find the moment generating function (MGF) of $T$. This appears to ...
3
votes
1answer
46 views

How to read the mean and standard deviation from a Q-Q plot

I wrote the following code in R to make the following qqplot. In a question I am asked to read the mean and standard deviation of the sample off the plot. I see ...
-1
votes
0answers
17 views

Hypotheses testing problem (MCQ) [on hold]

Please refer to the question in this image (more than one option may be correct): According to the answer key, the correct options are 1 and 2. I want to know what is the reason for not including ...
5
votes
1answer
65 views

Convergence in probability for a given distribution

$X_n$ assumes the values $1 , \frac{1}{2} , \frac{1}{3}, \ldots, \frac{1}{n}$. with equal probability . Examine whether $X_n$ converges in probability to some constant? This is what I tried : $\...
0
votes
0answers
55 views

iid random variables [duplicate]

Let X,Y,Z be independent, identically distributed random variables,each with density $f(x)=6x^5$ for $0\leq x\leq1$ and 0 elsewhere.I want to find the distribution and density functions of the maximum ...
-1
votes
0answers
30 views

Symmetric distribution problem (MCQ)

please refer to the question in image source :CSIR NET Mathematical science exam there is only one correct option and answer key says its option 4. since $X_1$ is symmetric so $X_1$ and $-X_1$ ...
3
votes
1answer
52 views

Need help to understand Feller's statement “whenever $r$th moment exists so do all preceding moments”

I am reading the book of Feller called "An Introduction to Probability Theory and Its Applications, Vol I" (third edition, page 227) and am stuck at the moment he explains the notion of variance of a ...
0
votes
1answer
54 views

Another round of some simple Bayesian probability questions

Following questions like this one, I have some very simple probability questions for which I guess that Bayes theorem is the tool to answer. I think that Bayes theorem is the tool because each ...
0
votes
1answer
133 views

Probability and Counting

Suppose You are ordering two pizzas.A pizza can be small,medium,large or extra large,with any combination of 8 possible toppings( getting no toppings is allowed, as is getting all 8) How many ...
0
votes
1answer
64 views

Sufficient statistics from exponential distributions with different means

If $X$ and $Y$ are independent exponential random variables with means $\theta$ and $2\theta$ respectively, then show that $X + 2Y$ is sufficient for $\theta$. I know how to find sufficient ...
-1
votes
0answers
41 views

Obtaining conditional density while sampling from exponential distribution

Given : $X$ and $Y$ are independent exponential random variables with means $\theta$ and $2\theta$ respectively, the conditional density for $(X,Y)\mid X + 2Y $ is desired problem in attempt : I ...
1
vote
1answer
310 views

Computing Partial Dependence Plots for Trees

Disclaimer: Although the Partial Dependence Plot derives from a textbook, and the method in question is mentioned in an exercise, I'm not trying to solve this for a class or homework assignment. I'm ...
-1
votes
0answers
11 views

M/M/1 Queue problem [closed]

Let $X(t)$ be the number of customers in an M/M/1 queuing system with poisson arrival rate $\lambda$ > 0 and service rate $\mu$ >0 it is known that $\lim_{n \to \infty}P(X(t)=1)=\frac{1}{4}$ then ...
7
votes
5answers
1k views

Random variable defined as A with 50% chance and B with 50% chance

Note: this is a homework problem so please don't give me the whole answer! I have two variables, A and B, with normal distributions (means and variances are known). Suppose C is defined as A with 50% ...
7
votes
2answers
973 views

Elements of Statistical Learning alternatives

Elements of Statistical Learning (ESL) is a book that has fantastic breadth and depth. It covers the essentials to the very modern methods by citing the papers where these original studies come about. ...
0
votes
1answer
22 views

Degrees of freedom for error in linear regression model [if n < k]

Given a multiple linear regression model with n= 3 observations and k= 4 regressors what will be the degrees of freedom for error sum of square ?
1
vote
1answer
265 views

Distribution of combination of events with different distributions themselves

I am asked to find the PMF of $X$, for the following definition of $X$: "There are two coins, one with probability $p_1$ of Heads and the other with probability $p_2$ of Heads. One of the coins is ...
0
votes
0answers
19 views

Bias of ROC curve

I am trying to use ROC curve with nonparmetric technique ($ROC_m)$. but 'm using following estimates of $\hat f$ and $\hat g$.$$\hat f=\frac {e^ -\frac{(x-t)}{√(h_x)}}{√(h_x ) \left(1+e^ -\frac{(x-t)}{...
3
votes
0answers
36 views

The minimal sufficient statistic of $f(x) = e^{-(x-\theta)}e^{-e^{-(x-\theta)}}$

The Casella Berger (2002) solutions manual says that the minimal sufficient statistic for $$f(x) = e^{-(x-\theta)}e^{-e^{-(x-\theta)}}, \qquad x\in \mathbb{R}$$ are the order statistics $(X_{(1)},\...
17
votes
2answers
1k views

Advantages of the Exponential Family: why should we study it and use it?

So here I am studying inference. I would like that someone could enumerate the advantages of the exponential family. By exponential family, I mean the distributions which are given as \begin{align*} f(...
5
votes
1answer
2k views

AIC equivalent to Mallows' Cp and Mallows' Cp unbiased for test MSE

Part 1: The goal is to show that with Gaussian errors and a linear model, Mallows' $C_p$ and $AIC$ are equivalent. Using our definition of Mallows' $C_p$: $$C_p=1/n(RSS+2d\hat\sigma^2)$$ and $AIC$: $$...
3
votes
1answer
58 views

First difference of AR(1) process

Given AR(1): $$X_t - \mu = \phi(X_{t-1}-\mu) + \epsilon_t$$ where $$ \mu = 0.85 \\ \phi=0.59 $$ and $$ W_t = X_t - X_{t-1} $$ Compute $$ Corr(W_t,W_{t-1})=-0.205 \\ Cov(W_t,W_{t-4})=-0.43 \\ Corr(...
0
votes
2answers
34 views

Variance of linear combination of AR(1) process

Let $ \{X_t\}$ ~ AR(1): $$ X_t=2.62-0.84X_{t-1}+\epsilon_t, \ \ \ \epsilon_t\sim WN(0,2.27)$$ Compute the variance of $$ \overline{X}= \frac{1}{3}\sum_{t=1}^{3} X_t $$ The solution is: Var($\...
0
votes
1answer
47 views

How to select the most important features in a KNN model, categorical & numerical data [on hold]

I need to find out which factors are the most relevant when predicting if a baby has a low birth weight, yes or no. My model looks like this at the moment: ...
-1
votes
0answers
50 views

OLS basic doubt

In a multivariate OLS model, $$ Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \epsilon$$ Is my estimator for $\beta_1$ given by: $$\hat \beta_1 = [X_1'X_1]^{-1} X_1'Y$$ or: $$\hat \beta_1 = [X_1' ...
3
votes
2answers
1k views

Approximating 1D integral with Metropolis - Hastings Markov Chain Monte Carlo

I've been asked to approximate the integral of a one dimensional unnormalised posterior with a flat prior, using a Metropolis Hastings Markov Chain Monte Carlo, I realise that this isn't a practical ...
3
votes
2answers
110 views

Importance sampling estimation of power function

Problem Suppose we are given $\text{Poisson}(\theta)$ model, and the null hypothesis is as follows: $$ H_0 : \theta = 0.1 \ \ \text{vs}. \ \ H_1 : \theta < 0.1 $$ Suppose we take sample of $n=...
0
votes
0answers
33 views

Maximum likelihood estimator for variance

Consider the linear model $Y\sim N_{n}(X\beta , \sigma^{2}I )$, where $X$ is a $n \times (k + 1)$ matrix of rank $k + 1 < n$. Let $\widehat{\beta}$ and $ \widehat{\sigma^2} $ be the maximum ...
2
votes
0answers
29 views

Asymptotic normality of quadratic form?

Let $X$ be a $p$-dimensional vector that is asymptotically normal such that $$\sqrt{n}(X - \mu_X) \stackrel{d}\longrightarrow N(0, \Sigma)$$, and let $H$ be a random $p\times p$ symmetric matrix, ...
0
votes
0answers
26 views

Pupils - Correlation between exam mark and answer from 1 to 5 of a test

100 pupils have their exam mark from 1 t0 50 and separately they have reposed a questionary with 10 questions. The variable that represents the answers for the questions of the questionary are ...
4
votes
2answers
3k views

Difference between contextual anomaly and collective anomaly

Contextual and collective anomalies are defined as follows (source): Contextual Anomalies. If a data instance is anomalous in a specific context (but not otherwise), then it is termed as a contextual ...
1
vote
0answers
34 views

Prove that the Pitman estimator is itself complete sufficient

It's a homework question, but I just have no idea about it ... Let be random variables according to a distribution having joint density ,where is a location parameter. Assume that there exists a ...
0
votes
1answer
22 views

Hypothesis testing of sum of uniform distribution

I want to have some clarifications related to below question from Casella Berger Regarding $\phi_1(X_1),$ I understand that $\alpha$-value is 0.05 and hence we are looking for a value of $C$ for ...
3
votes
1answer
33 views

solve an exercise of two samples using Kolmogorov-smirnov

I'm looking for books and information like crazy and I can not find what I need. Well the example proposed is about methods that have been used in literature students and these are the data collected: ...
4
votes
0answers
168 views

Recreating figure from Elements of Statistical Learning [closed]

I am trying to recreate FIGURE 3.6 from Elements of Statistical Learning. The only information about the figure is included in the caption. To recreate the forward stepwise line my process is as ...