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0
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0answers
26 views

$\text{cov}$$(\hat{y_i}, \hat{y_i}^*)$ follow-up

Consider the model: $ {y_i}= {\beta_0} + {\beta_1}x_i + \epsilon_i$, where $E[\epsilon_i]=0$ and $var[\epsilon_i] = \sigma^2$ for $i = 1, 2, ..., n.$ Let $\hat{\beta_0}$ and $\hat{\beta_1}$ be the ...
0
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0answers
26 views

What is ${\rm cov}(e_i, \hat y_i)$ in simple linear regression?

The model is $y_i = \beta_0 + \beta_1x_i + \epsilon_i$ What is ${\rm cov}(e_i, \hat y_i)$? What is ${\rm cov}(\epsilon_i, \hat \beta_1)$? What is ${\rm cov}(e_i, \epsilon_i)$? For 1, I am writing ...
-3
votes
0answers
21 views

Fitted coefficients [on hold]

(b) The fitted coefficients and summary table including t-statistics and the overall F statistic for the fit etc. (c) Appropriate diagnostic plots. (d) Description of any action taken, as a result ...
-1
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0answers
26 views

Diagnostic plot includes outliers

Write a short report of your analysis which should include; (a) Plots of the response versus each of the predictors. (b) The fitted coefficients and summary table including t-statistics and the ...
0
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0answers
21 views

Probability of Event from a Chi-Square Table

I have the following table: From here I want to answer the following question: P(CHILD CAN RECOG BEG SOUNDS OF WORDS – YES | CHILD RECOGNIZES LETTERS - Most of them). From my understanding of ...
7
votes
2answers
229 views

Functions of Independent Random Variables

Is the claim that functions of independent random variables are themselves independent, true? I have seen that result often used implicitly in some proofs, for example in the proof of independence ...
1
vote
1answer
40 views

Taylor approximation to penalty function

I'm reading through Fan's SCAD paper and I feel like I'm not getting a simple step. On page 1354 where he is talking about quadratic approximations to a penalty function he has ...
6
votes
1answer
107 views

Gibbs sampling for Ising model

Homework question: Consider the 1-d Ising model. Let $x = (x_1,...x_d)$. $x_i$ is either -1 or +1 $\pi(x) \propto e^{\sum_{i=1}^{39}x_ix_{i+1}}$ Design a gibbs sampling algorithm to generate ...
0
votes
1answer
21 views

Probability problem regarding rooks on a chessboard

Eight rooks are placed in distinct squares of an 8 x 8 chessboard, with all possible replacements being equally likely. Find the probability that all the rooks are safe from one another.
1
vote
1answer
25 views

Naming of dependent and independent variables in simple linear regression

I'm currently doing some self-study on Simple Linear Regression. I came across the terms "dependent variable" (eg. Sales volume) and "independent variable" (eg. Advertising budget). I am curious to ...
0
votes
1answer
25 views

Combining results of two samples to find combined standard deviation

I was making an attempt to solve the following problem sets, however, my answer 15.913 seemed quite different from that in the answer key 16.84. I have checked my answer a few times and re-doing the ...
0
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0answers
14 views

Finding P Value of Two sample hypotesis test for Proportion

I am attempting to find the P-value of a hypothesis test, which compares the success rate of two procedures. A test with 0.05 level of significance is conducted to test the success rate of two ...
1
vote
1answer
28 views

Exponential Distribution Expected Value

Question 1 (part 1) I was attempting to use the exponential distribution to solve the following problem: the time between cars passing on a road can be modelled with an avg rate of 3.5 cars per ...
3
votes
1answer
214 views

Newton's algorithm for a Minimax Procedure

In order to find the minimax solution of a certain problem I have to determine the constant $c$ that provides the root for the following equation: $$l(c)=3\times \left [ 1-\Phi \left( c-75 \right) ...
1
vote
1answer
15 views

IQR for Normal Probability Distribution

I was attempting to find the IQR for a normal distribution with mean being 4.5, while the standard deviation. My attempt yielded the result 2.144, while the actual answer provided was 2.1584. ...
-2
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0answers
13 views

group differences [on hold]

Assume that you have been provided with data on 12 conceptually distinct variables that are moderately correlated with one another at the empirical level. The data are from students in four different ...
2
votes
1answer
53 views

Self-Study question on Normal Distribution

I was attempting a self-study question on the normal distribution, but I wasn't able to get to the answer. I'm not sure if my method used here is incorrect, and would appreciate some guidance and ...
2
votes
1answer
59 views

Question on Probability

I was attempting to solve a question on finding the probability that a randomly selected person had an income of $50,000 or more would be audited, based on information on the table below: My ...
1
vote
0answers
31 views

Calculation of standard variance $s^2$

First picture is the question and its answer key: The second picture is my solution: I know that $S^2=\frac{\sum(x_i-\bar x)^2}{n-1}$ But in the answer key, $s^2$ cannot be found in that way. ...
0
votes
2answers
27 views

Hypergeometric Distribution Question

I am currently doing a self-study question that gave me the following scenario: There are 10 marbles in a jar. 7 marbles are red while 3 are blue. John wants to pick 4 marbles out to give his friend. ...
1
vote
2answers
70 views

How to produce the minimum forecast error using R?

Considering that we want to use optimize() on the interval [0,1] how can I write an R code for finding the value of β that produces the minimum forecast error without using external packages like ...
3
votes
0answers
28 views

Correlation & Regression Prediction [on hold]

I have a homework question. I have solved most of it already, but am unsure how to proceed with one specific part that involves prediction (Parts B & C). I am not looking for anyone to just give ...
3
votes
2answers
139 views

PDF of Sum of Two Random Variables [on hold]

$X$ and $Y$ are uniformly distributed on the unit disk. Thus, $f_{X,Y}(x,y) = \begin{cases} \frac{1}{\pi}, & \text{if} ~ x^2+y^2 \leq 1,\\ 0, &\text{otherwise.}\end{cases}$ If $Z=X+Y$, find ...
2
votes
2answers
40 views

Estimation of standard error

I was doing some self study and came across the following formulae for estimating standard errors: Formulae 1: Formulae 2: I understand that these two can all be used when the Population ...
4
votes
1answer
94 views

Using central limit theorem for approximation

Let $X$ be a random varaible from a distribution with pdf $$ f(x) = \theta x^{\theta-1}, \quad 0< x < 1. $$ a) Name the distribution of $U=-\ln(X)$ by first finding its density ...
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0answers
14 views

Dynamic Regression

Problem Suppose that a typical firm determines its level of stocks $H_t$, in accordance with the following rule: $H_t - H_{t-1} = \lambda (H^*_t - H_{t-1}) + \epsilon _t$ where $\epsilon _t$ is a ...
8
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6answers
159 views

Reference Request: Generalized Linear Models

I am looking for an introductory to intermediate level book on Generalized Linear Models. Ideally, in addition to the theory behind the models, I would want it to include applications and examples in ...
1
vote
1answer
32 views

When to use CDF and PDF for Exponential Distribution

I am doing a self-study on Exponential Distribution and noted an exponential on my text giving that: CDF of Exponential Distribution $$ F(x) = 1 - e^{-λx} , $$ PDF of Exponential Distribution $$ f(x) ...
1
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1answer
30 views

Continuity correction error when using normal distribution to estimate Poisson distribution

Good morning Everyone, I am doing a self-study exercise which attempts to exemplify a case where the Normal Distribution is used to approximate the Poisson Distribution, since the population mean is ...
1
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0answers
28 views

Finding Percentage Point of a Normal Distribution

Good evening all, I am doing a self-study exercise, but have been quizzed by a part of the question on finding percentage points of a normal distribution. I fully understand the first part of the ...
1
vote
1answer
120 views

What does the | in |Z| mean in mathematical expressions for distribution statistics

I am doing some self study on statistics and noticed that in the notes that I was using the $|Z|$ expression as attached in the photos below. I am confused with the "$|$" that is being used. The only ...
0
votes
0answers
36 views

How does this algebraic relationship among expectations work?

Find the expected mean squares error of lack of fit. Trial: $$SSLOF=\sum_{1}^mn_i(\bar y_i-\hat y_i)^2\\=\sum_{1}^mn_i(\bar y_i-\bar y)^2-\sum_{1}^mn_i \hat\beta_i^2(x_i-\bar x)^2$$ and ...
1
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0answers
19 views

What Gaussian process covariance function captures an affine mean?

In the documentation for GPML, the author trained a GP with an affine mean function and isotropic squared exponential covariance function. Then there is an exercise to the reader: Try training a ...
0
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0answers
17 views

Compare the marginal effect for a dummy variable with a continuous variable from a probit model

I use probit model for one regression which consists one independent dummy variable that captures whether the firm is unionized or not. I also estimate the same probit model instead of consisting ...
0
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0answers
82 views

2 sided vs 1 sided test

I am asking this questions straight from "Data Analysis and Statistical Inference " course by Dr. Mine Çetinkaya-Rundel on coursera! Please right click on the image and one in a new tab to make it ...
0
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0answers
21 views

Bayesian mean square error

Given a i.i.d sample $X_{1},..,X_{n}$ of bernoulli random variables test 2 hypotheses $H_{0}:p=2/3$ and $H_{1}:p=1/3$. Bayesian prior is $\pi(2/3)=1/3$ and $\pi(1/3)=2/3$. Find the bayesian criterion ...
2
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0answers
25 views

How would I find the P-Value of this data?

I have the following data (relating to consecutive months at current job): Mean: 45.4 Standard Deviation: 60.89 Sample Size: 48 I am asked to do a one-tailed test at a 1% significance level and ...
2
votes
1answer
63 views

How to compare whether the coefficients of two independent variables statistically different from each other?

If I have two independent variables and they are dummy variable along with other independent variables and I run a linear probability model, I want to compare whether the coefficients of two dummy ...
1
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0answers
45 views

Poisson distribution confidence interval

Edit : this is the data I used for the first part of the problem : \begin{matrix} Rocks & 0& 1& 2& 3& 4& 5& 6 & 7\\ samples & 12& 27& 28& 19& ...
0
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0answers
18 views

White standard errors and Weighted Least Squares (WLS) [closed]

Are there situations in which it would be useful to apply WLS and use White standard errors (from the transformed model) as well?
2
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0answers
32 views

Which observation has the largest variance of the residual?

a) For which of the following observations (obs1, obs2, obs3) is the variance of the residual the largest? Which observation has the highest leverage? And which one the smallest? Explain why. ...
1
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0answers
20 views

How to calculate false accept and reject rates in pattern recognition?

I am working on a vein pattern recognition project based on SURF algorithm and Euclidean distance. I have completed my program to find the maximum and minimum distance between vein features and find a ...
2
votes
1answer
45 views

Conditional Probability equal to zero

Let $X_1$ and $X_2$ be two independent Normal variables with mean $\theta$ and variance $1$. Let $S=X_1+X_2$ and consider the following conditional probability: $$P \left[ X_1 \leq x_1; X_2 \leq x_2 ...
0
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0answers
31 views

Proving that Markov Chain Monte Carlo converges

I actually asked the same question in http://math.stackexchange.com/ as well at http://math.stackexchange.com/questions/753105/proving-that-markov-chain-monte-carlo-converges but since the question is ...
2
votes
2answers
49 views

Finding a statistically different proportion

I have the following problem: given a proportion $p_1=\frac{X_1}{n}$, where $X_1$ is the number of succesful outcomes in $n$ tries ($X_1$ and $n$ are both > 100), I have to find the smallest ...
1
vote
1answer
42 views

How should you fit ANOVA and linear regression models, if the equal variance assumption is violated?

This is my topic for the paper I'm working on for an undergrad stats class. It's supposed to be 20 pages... and I'll be honest, I understand very little beyond the basics and am over my head. From ...
1
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0answers
24 views

Expected mean square of random effect in CRD model

Random effect in CRD model: \begin{align} y_{ij} &= \mu + \tau_{i} + \varepsilon_{ij} \\ \tau_{i} &\sim \mathcal N(0,\sigma_{\tau}^2) \\ \varepsilon_{ij} &\sim ...
4
votes
2answers
244 views

Expected sum of money

There is a little problem in the working on this one I have encountered recently. Here it goes: Starting off with \$100, when you toss a coin, if there is a tail, you will get \$10 and lose $10 if it ...
1
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0answers
18 views

Meaning of marginal prior distribution

I have a variable $\mu$ with distribution, given $\phi$ of $ \mu|\phi \sim N(\phi, \sigma^2)$, and the distribution of $\phi$ is $\phi \sim N(\nu, \tau^2)$. (Here, $\sigma^2,\nu$ and $\tau^2$ are ...
0
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0answers
12 views

Choosing an appropriate proposal distribution for metropolis hastings

So far the only constraint I've found for sampling from some target distribution $\pi(x)$ is that the proposal distribution must include the support of $\pi(x)$. That's very vague. What makes a ...