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16 views

On conditional/joint probability [duplicate]

Problem 1 The table on the left shows the joint probability distribution between two random variables - X and Y; and the table on the right shows the joint probability distribution between two random ...
1
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0answers
15 views

How to prove that Radial Basis Function can be derived by mapping function?

How to prove the radial basis function $k(u,v) = \int_{\mathbb{R}^d} \phi_t(u)\phi_t(v)dt $ can be integrated out by mapping function? $$\phi_{t}(u) = \frac{1}{(2\pi\Sigma)^{d/2}} ...
2
votes
1answer
57 views

Expected value of Y = (1/X) where $X \sim Gamma$

I'm having some confusion over this statement here. Let $T_i \sim Exp(\lambda + \theta)$ and if they are all iid then $\sum_n T_i \sim Gamma(\alpha = n, \beta = 1/(\lambda + \theta))$ I want to find ...
0
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0answers
19 views

Exponential distribution for lightbulbs

The lifetime of a certain type of a lightbulb is 20 months. If a household has 12 light fixtures whose lightbulbs are replaced at the same time. Determine the probability that 5 of the lightbulbs will ...
0
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1answer
18 views

Calculating Standard Deviation when given sample size, mean difference & p value

I am trying to pool data in my meta-analysis and i need MEAN & SD. However the study has reported sample size (27), before (11.8) & after mean (11.9), and p value (0.540). I need the SD. ...
1
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1answer
23 views

Posterior predictive test quantities

I've been trying to figure out problem 6.2 from Gelman's book, second edition, page 192 on Bayesian data analysis. Can anyone help? a) Set up predictive test quantities to check the following ...
0
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1answer
37 views

what should be the null hypothesis?

conduct a test hypothesis to show that it takes longer for people with average fitness to successfully complete physical therapy as compared to those with above average physical fitness. Test at ...
0
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1answer
18 views

Statistical independence of least square estimator and residual in multiple linear regression

I'm currently self studying linear regression. Following is an entrance exam problem of a graduate school. Consider the regression model with usual assumptions of the errors $y=X\beta+\epsilon$. Show ...
0
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0answers
28 views

Exponential distribution of the time between recurrences

Patients in a hospital are treated for certain illness. The time in days that no recurrence takes place follows exponential distribution with mean of $\theta = 27$ days. The following table gives the ...
0
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0answers
6 views

computing expectations in variational updates

I have a complete log-likelihood expression as follows: $$ L = \sum_{i=1}^N \log P(y_i|x_i, w_i, \beta) + \log P(\beta) + \sum_{i=1}^N \log P(w_i) $$ Now, I need to compute the expectation of these ...
0
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0answers
10 views

Econometrics 2: Covariance (xt, xt+1) in Moving Average process of order 1,

Could you help me to show that if X(t+1)=e(t+1)+alpha(1)*e(t), Then Cov[x(t), x(t+1)]=alpha(1)*Var[e(t)] Thank you very much, your help is appreciated
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0answers
21 views

t-student distribution [duplicate]

I've got this problem: Here, if $Z,W$ are independent random variables, and $Z$ has normal standart distribution and $W$ has $\chi^2$ with $n$ degrees of freedom, $T=\frac{Z}{\sqrt{\frac{W}{n}}}$. I ...
2
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0answers
31 views

Asymptotic distribution of $\hat{B_1}$in simple linear regression

I am currently studying how to $\bf{identify}$ the parameter $B_1$ in a simple univariate regression model where we have $Y=B_0+B_1X+\epsilon$ with the usual assumption of $X$ being exogenous, ...
1
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1answer
36 views

Cook's D, testing for outliers

I am working on a multiple linear regression and I want to check for outliers using Cook's D. I have a problem interpreting it, as there are many points above the 4/N line, but only one is >1. How ...
0
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0answers
10 views

How can I prove the variance of residuals for an unbalanced one-way layout?

How can I prove the variance of residuals for an unbalanced one-way layout? var($e_{ij}$)=$σ^2$($1- \frac{1}{n_i})$ here i = 1,...., a j = 1,....$n_i$ where $e_{ij}$'s are the residuals. where ...
1
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0answers
39 views

Calculating $E\big((X_1X_2)/(\sum_{i=1}^n X_i^2)\big)$ [on hold]

Let $X_1,X_2,...,X_n$ be a random sample of $N(0,\sigma^2)$. How can I calculate $$ E\bigg(\frac{X_1X_2}{\sum_{i=1}^n X_i^2}\bigg)\ ? $$
0
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0answers
192 views

Elementary counting arguments: Family with 6 children, gender and birth

Problem 2.1.5 from Basic Concepts of Probability and Statistics, 2nd Ed. by Hodges and Lehmann Suppose you know that a family has 6 children, 2 boys and 4 girls, but you don't know their sexes ...
1
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1answer
27 views

Stationarity of AR(1) process whose autoregressive parameter could change over time

Imagine an AR(1) has an autoregressive parameter which could change in time. $y_t-\mu=\phi_t (y_{t-1}-\mu)+\varepsilon_t\,$, where $\phi_t$ is not always constant but still lies inside the usual ...
2
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3answers
193 views

Confused on the interpretation of regression coefficients

Let's suppose we have the following regression model: $$Y_i=\beta_0+\beta_1D_i+\beta_2D_iX_i+\epsilon_i$$ where $Y_i$ represents the test score of the i-th student, $D_i$ is a dummy variable that ...
0
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1answer
70 views

Mathematical Modeling and Statistical Modeling

What is the difference between mathematical modeling and statistical modeling? I only know that a mathematical model is deterministic while a statistical model is stochastic. Is that all to answer ...
2
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1answer
47 views

How do I generate two correlated Poisson random variables?

How would I simulate observations from a bivariate Poisson distribution such that they have a nonzero covariance? The hint I was given is that I need to use the fact that the sum of two Poisson random ...
2
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1answer
39 views

On an implication of the memoryless property of the exponential random variable

I know that if we take $X \sim Exp(k)$ then we have this property: $$P(X \ge s + t | X \ge s) = P(X \ge t)$$ But why does this imply that $X | X > x$ has the same distribution of $X$ only ...
5
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3answers
162 views

Visualizing variability from graph

It is written in the book Applied Logistic Regression, Second Edition. By David W. Hosmer and Stanley Lemeshow , p.2 that a problem with the above graph is that the ...
0
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1answer
26 views

No ARIMA, No GARCH, which model?

I am trying to fit a model for a data set. The acf and pacf, after differentiating the data are: The acf shows that the returns appeared to be random. According to them seems like the ARIMA model ...
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2answers
24 views

Observed and Observable

What is observed and what is observable? I found this two word frequently in the context of random variable and realization of ...
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0answers
7 views

References regarding correlations with ranks and binary data

I have several rank variables (ranks 0-3), which can be reasonably turned into binary (significant/insignificant effect). I'm looking for potential interactions. What would be the best source to look ...
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0answers
35 views

Dear σ2, please, be small and constant! What do you mean – “NO”? [closed]

pls can someone help me, i have project about "Dear σ2, please, be small and constant! What do you mean – “NO”???" can someone help me and suggets how to approach this topic and what book or ...
0
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1answer
12 views

Total Variation Denoising help

I am trying to work through the "Mathematical exposition for 1D digital signals" in the wikipedia entry for Total Variation Denoising (TVD). I am familiar with Lagrange multipliers. However, I cant ...
3
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3answers
130 views

How to test if my distribution is multimodal?

When I plot a histogram of my data, it has two peaks: Does that mean a potential multi-modal distribution? I ran the dip.test in R (...
0
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0answers
24 views

MLE of k given $p=1/2$ in Binomial(k,p) [closed]

Let $X_1,...,X_n$ be a random sample from a binomial(k,p) distribution. If $p=1/2$, how to proceed to find the MLE of k? The obvious procedure of maximizing the likelihood function is not yielding ...
0
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0answers
10 views

Conjugate Distributions

I read one way to derive saddle point approximation uses the so called conjugate distributions densities $h_t(x)$ for a given density function $f(x)$. It is defined as follows. $$ h_t(x) := ...
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1answer
34 views

Show posterior is proper for this poisson linear model

This question is 3.12 in Andrew Gelman's Bayesian Data Analysis 3rd edition. Let $y_i|\alpha,\beta \overset{iid}{\sim} \text{Poisson}$ with mean $\alpha+\beta t_i$. Find a prior distribution that ...
0
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1answer
53 views

How multicolinearity affect the prediction

In a linear regression model, if some of the predictors are correlated, then in the output of most software, you will see very large p-value in those coefficients and very high standard error. My ...
0
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1answer
36 views

What does “if” mean in this question?

What is the probability that Sam is guilty if Tom and Devi gave conflicting testimonies? Is it conditional probability? Or intersection simply?
1
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1answer
32 views

Simple question about finding conditional expectation

Let $X,Y \sim U[0,1]$ ($X,Y$ are independent), we want to find $E[X|X>Y].$ I tried a few approaches to the above problem, but am not confident in my answer. One approach is as follows. Note that ...
1
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1answer
43 views

Permutation & Probability

There are total of 10 persons in a queue. If two particular people out of the 10 persons are to be together, what is the probability of that happening? Do I group the 2 people together and then insert ...
1
vote
1answer
38 views

Sufficient statistic for a Gamma distribution

I am confused about the steps I need in order to solve the equation below. I must use conditional distribution (and NOT the factorization theorem). Q: $X_1, . . . , X_n$ is a random sample from a ...
1
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0answers
16 views

On Kolmogorov's Theorem In Time series theory and methods (1990)

I am following Time series theory and methods, Brokwell and Davis (1990). And theorem 1.2.1 called by the text Kolmogorov's Theorem is only stated but not proven. I will rewrite it here: The ...
3
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1answer
46 views

How to make sample space of the following question

My teacher gave me a question to solve.In the book of Ronald E. Walpole (Probability and Statistics for Engineers and Scientists), chapter no#3 exercise 1,question #2 which is as below: 3.2) An ...
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0answers
8 views

Acceptance Sampling Plan

I was reading Acceptance Sampling Plan but didn't understood when should I use single sampling plan , when double sampling plan , when multiple sampling plan and when sequential sampling plan. Also ...
2
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1answer
52 views

Non-robustness of parametric statistics

Why is parametric test considered to be non-robust ? Or, why is parametric test not considered to be robust?
2
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1answer
24 views

Parameters in a non-parametric model

I have not understood this Wikipedia statement: The difference between parametric model and non-parametric model is that the former has a fixed number of parameters, while the latter grows the ...
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0answers
15 views

finding UMVUE of parameter $\gamma(\theta)=P_\theta(X_1+X_2\leq a)$ [closed]

Let $X_1,X_2,...,X_n$ be a random sample of $N(\theta,1)$. how can find UMVUE of parameter $\gamma(\theta)=P_\theta(X_1+X_2\leq a)$ where $a$ is Constant and known
5
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2answers
73 views

What information can be retrieved from the slope in linear regression

Here is the problem: The question is weird in the fact that we're using categorical variables, but I see it this way: Since one extra year of education has 4 times the effect on income as going ...
0
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0answers
13 views

Conceptual Question on Gaussian Mixture model in higher dimesnion

Consider a smooth manifold $M=R^d$ embedded in a higher dimensional space $R^D$ using Takens Attractor reconstruction. Let, the Random Variable $X \in R^d$ have a Gaussian pdf and the random variable ...
1
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0answers
55 views

PCA and diagonalization of the covariance matrix

I am preparing for an upcoming exam and having looked at older exams I found one PCA related question I am having trouble understanding: You have a dataset of $N$ two-dimensional points $\ y^t $. ...
0
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1answer
30 views

Difference between “Design based approach” and “Model based approach”?

In a pdf file, i found the following thing which i have not understood at all. ‐ one view (e.g., Heckman, 2008): causality is model‐based: causality only exists within the framework of a theory ...
3
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0answers
19 views

Poisson convergences of series of random variable

Let $X_i$ be independent Poisson with mean $\lambda_i$. Show that i)$\sum_1^\infty\lambda_i<\infty$ implies $\sum_1^\infty X_i$ converges almost surely to a finite limit. ...
0
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0answers
9 views

Why are the least squares standard errors for the (unrestricted) ADL model are valid?

Consider the model $\Delta y_t =\beta_0+\beta_1 x_t+u_t$ where $u_t=\phi u_{t-1}+\varepsilon_t$; the $u_t$ are IID. If we write this in ADL form there are restrictions on the parameters. Why are the ...
2
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0answers
28 views

Expectation of cube of summation of independent random variables

Where would I begin on this problem? I know I begin with pulling $c^3$. Where would I go from there? And I know that $\mathbb{E}[X] = x_1p_1 + .... x_n p_n$ I'm stuck on the rest, however.