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3
votes
0answers
23 views

Correlation & Regression Prediction

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 ...
2
votes
2answers
64 views

PDF of Sum of Two Random Variables

$X$ 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
38 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
0answers
53 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 ...
0
votes
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 ...
5
votes
4answers
80 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
28 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
vote
1answer
24 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
vote
0answers
27 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
116 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
vote
0answers
15 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
votes
0answers
15 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
votes
0answers
77 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
votes
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
votes
0answers
22 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
61 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
vote
0answers
42 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
votes
0answers
18 views

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

Are there situations in which it would be useful to apply WLS and use White standard errors (from the transformed model) as well?
2
votes
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
vote
0answers
19 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
votes
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
46 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
vote
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
vote
0answers
58 views

What is the distribution of this ratio of quadratic forms?

I guess it's F-distribution but I don't know what the solving process it is
0
votes
0answers
22 views

Prove mutually independent

I tried it using some theorem...but I can't Can someone give a proof of this plase? I can't sleep because of it!
1
vote
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
votes
0answers
11 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 ...
2
votes
1answer
30 views

Representation of ARMA processes

Question Consider the following process: $$2y_t-3y_{t-1}+y_{t-2}=\epsilon_t-\theta\epsilon_{t-1}$$ What is the model for the process $w_t=\Delta y_t = y_t-y_{t-1}$? Attempt I have solved the ...
0
votes
1answer
31 views

Learning multivariate techniques to analyze the gut microbiome

I would like to start learning some of the basics regarding data processing and analysis of microbiome data using R. Can anyone recommend a tutorial on some of the core/basic approaches that also ...
0
votes
2answers
47 views

Expected value of multiplication of Identically distibuted random variables

i am trying to understand if the following statement is true: $$ E(XY)= E(X^2)=E(Y^2) $$ if $X$ and $Y$ are identically distributed but not necessarily independent r.v. This means that if the ...
1
vote
0answers
30 views

Marginal distribution of a function of order statistics

From the joint distribution of any two order statistics, say $Y_j$ and $Y_k$, $j<k$ I would like to derive the distribution of $Z=F(Y_k)-F(Y_j)$. The initial pdf is: $$f_{Y_j,Y_k} (y_j,y_k) ...
4
votes
1answer
79 views

Covariance term in simple linear regression

I am trying to derive the expression for the variance of $\hat{\beta_0}$ in simple linear regression. I substitute $\bar{y} - \hat{\beta_1} \bar{x}$ for $\hat \beta_0$, but in the intermediate steps ...
0
votes
0answers
11 views

Homogeneity of Variance, 2 Way Completely Crossed Design

I am looking for some advice as to how I might handle having an unfavorable Levene's test outcome, that is to say a highly significant value. DOE Various containers with water were microwaved for ...
0
votes
1answer
73 views

How to get P-Value when t value is less than 1?

I am doing an assignment for class and I have hit a wall. I'm not sure what to do next. Using a sample with 12 discrete elements, I am trying to test a hypothesis that the population mean is 243 ...
0
votes
2answers
45 views

Function of random variable

If I have a random variable $X$ which has mean $\mu$ and variance $\sigma^2$, what is the approximate expression of $log(X)$ and $\sqrt{X}$? Do I assume normal approximation or use Taylor expansion?
0
votes
0answers
23 views

Assuming training data as set of target funtions

In this 10th slide of http://www.cs.cmu.edu/afs/cs.cmu.edu/project/theo-20/www/mlbook/ch6.pdf presentation The Training data set $D$ is assumed as the set of target function . Actually $D = ...
0
votes
0answers
23 views

Standard mean and deviation

A company supplies specialized high-tensile pins to a customer.It uses an automatic lathe to produce the pins.Due to factors such as vibration, temperature and wear and tear, the lengths of pins are ...
2
votes
1answer
44 views

showing a random variable has an exponential distribution

Let $X_{1},..,X_{n}$ be independent, each with a exp($\lambda$) distribution. Let $Z=min(X_{1},..X_n)$. Show that $n\lambda Z$ has an exp$(1)$ distribution. I calculate that $P(Z>z)=e^{-n\lambda ...
1
vote
1answer
47 views

Formulas for probabilities in Bayes theorem

In continuation to this question $p(h|D) = \frac{p(D|h)p(h)}{p(D)}$ $p(h) = $prior probability of hypothesis $h$ $p(D)$ = prior probability of training data $D$ $p(h|D)$ = probability of $h$ ...
3
votes
1answer
54 views

Simulate the distribution of the median

I have done part(a); now I need to do part(b): I want to simulate the distribution of the median $M$ using R. I don't know how to work it out. And for part(c), do I need to use R as well? This is ...
1
vote
0answers
16 views

Gamma Distribution MGF

I'm not sure how to find part a, what formula can be used to find this mgf?
0
votes
0answers
34 views

Random model selection and validity of significance tests

Suppose we have some data, $\{y_i, x_{1i}, \dots, x_{ki}\}_{i=1}^n$ and we want to build a linear model of the form $y_i = \beta_0 + \beta_{1'i}x_{1'i} + \dots + \beta_{k'i}x_{k'i} + \epsilon_i$, ...
0
votes
0answers
44 views

Poisson distribution help

In many cities, neighborhood Crime Watch groups are formed in an attempt to reduce the amount of criminal activity. Suppose one neighborhood that has experienced an average of 10 crimes per year ...
1
vote
1answer
77 views

Constructing one-ninth fraction of the $3^{5}$ design

I want to construct a $3^{5-2}$ design with $I=ABC$ and $I=CDE$ as generators. The complete defining relation for this design is: $I=ABC=CDE=ABC^{2}DE=ABD^{2}E^{2}$. This is a resolution III design ...
5
votes
3answers
163 views

What is a “strictly positive distribution”?

I am reading Judea Pearl's "Causality" (second edition 2009) and in section 1.1.5 Conditional Independence and Graphoids, he states: The following is a (partial) list of properties satisfied by ...
0
votes
1answer
62 views

construct the maximum likelihood estimator

Let $a_{1},a_{2},a_{3}$ be independent with a normal(0,1) distribution. Define $X_{1},X_{2},X_{3}$ by $X_{1}=a_{1}$, $X_{2}=\theta X_{1}+a_{2}$ and $X_{3}=\theta X_{2}+a_{3}$ Find the MLE for $\theta$ ...