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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
1answer
66 views

variance of nonparametric estimator of mean

I'm having some trouble with understanding how to calculate the variance of a non-parametric estimator. The example comes from Wasserman's "All of statistics book" Let $X_1, \ldots,X_n \sim \text{...
1
vote
1answer
40 views

Distribution of a one realization of a stochastic process [closed]

Suppose $X$ is a stochastic process such that $X(t) \sim N\left(\mu(t), \sigma^2(t)\right)$ for all $t$ and $\mu$ and $\sigma$ are some smooth functions and we are given one realization of this ...
-1
votes
1answer
31 views

Estimation of an exponential parameter

I´m trying to figure out the pdf $f_\min(X_i)$ of $\min(X_i)$, where the distribution of the sample $X_1,...,X_n$ is $\mathcal{E}xp(\lambda)$, where $\lambda$ is the unknown parameter. I tried with ...
0
votes
0answers
32 views

Order statistics of a single r.v

I have trouble understanding the following question: We have N i.i.d random variables from the uniform distribution between 0 and 1. If N=1, what is the probability that the n^th order statistic ...
9
votes
2answers
488 views

Probability of $X_1 \geq X_2$

Suppose $X_1$ and $X_2$ are independent geometric random variables with parameter $p$. What is the probability that $X_1 \geq X_2$? I am confused about this question because we aren't told anything ...
1
vote
0answers
35 views

The Linear Discriminant Analysis Rule

Given there are two classes A and B and the prior probability of belonging to $ A = Na/N $ and $B = Nb/N $, I want to show that the linear discriminant analysis rule classifies an observation x to ...
1
vote
0answers
29 views

When is the pmf of the difference of two independent random variables symmetric in zero?

Consider the stepwise cumulative distribution function $$ \Delta(x; \lambda, \mu)=\sum_{j=1}^J \lambda_j 1\{x\geq \mu_j\} \hspace{1cm} \forall x \in \mathbb{R} $$ where $J<\infty$ $\lambda\equiv (\...
4
votes
0answers
29 views

Partitioned regression model: estimator of beta 1

below is an exercise that is really giving me a hard time, I believe that there is a simple way around it but I can not find it: Assume the correct regression model is Y = X$\beta$ + $\epsilon$ for E(...
2
votes
2answers
39 views

How to change in y variable having means, standard deviations, and correlation coefficient?

The following statistics have been obtained by using the data of money supply growth rate and inflation rate for the last 30 years. The average money supply growth rate, its standard deviation,...
0
votes
0answers
11 views

Formula for Power of Upper Tail Sign Test

Given Consider the upper tail $H_0: \theta \leq 0 \,\,\, \text{vs} \,\,\, H_A: \theta > 0$ sign test with the test statistic $B = \sum_{i=1}^n{\psi_i}$ where $\psi_i = \mathbb{I}(Z_i > \theta)$...
-3
votes
1answer
36 views

Statistics and Probability. Please show solution [closed]

Miss Romero noted that the mean scores of a random sample of 15 grade 8 students who had taken a special test were 80.5. If the standard deviation of the scores was 3.1 and the sample came from an ...
0
votes
1answer
31 views

Odds Ratio Vs. Risk Ratio

Relative risk, odds ratio, risk ratio, risk difference - these are all measures of the direction and the strength of the association between two categorical variables. Can I use any of these four ...
0
votes
1answer
30 views

Constant Variance Assumption in Linear Regression

It seems to me that the following plot of "Residuals Vs. Fitted Values" violates the assumption of constant variance, since for lower fitted values, there are fewer points whereas for higher fitted ...
0
votes
1answer
34 views

Simple Regression Question for Probability of Smoking

Variable smokes is a binary variable equal to one if a person smokes and zero otherwise. We estimate a linear probability model for smokes: $$\hat{smokes} = .656 - .069\log(cigprice) + .012\log(...
0
votes
1answer
34 views

Properties of Nonparametric Test Statistics $B$

Given Consider the upper tail $H_0: \theta \leq 0 \,\,\, \text{vs} \,\,\, H_a: \theta > 0$ sign test with the test statistic $B = \sum_{i=1}^n{\psi_i}$ where $\psi_i = \mathbb{I}(Z_i > \theta)$...
0
votes
0answers
13 views

Prior for precision tau on Normal distribution with unknown mean [duplicate]

I am working with prior distributions. I need to define a prior for tau precision $\tau=1/\sigma^2$ for a Normal distribution with unknown mean $\mu$. Likelihood for normal distribution below tau ...
0
votes
1answer
23 views

Help with PCA Question

The conventional model for probabilistic principal component analysis has a standard normal latent $\vec{y}$ and a loading matrix $\Lambda$: $P(\vec{y}) \sim N(\vec{0}, I)$, $P(\vec{x}|\vec{y}) \sim ...
0
votes
0answers
68 views

MLE of Parameters of Bivariate Normal Distribution

I am working through find the maximum likelihood estimators of the bivariate normal distribution, without using matrices. I have the following density function: $f(Y_1,Y_2) = \frac{1}{2\pi\sigma_1\...
3
votes
1answer
136 views

Does UMVUE of $\frac{\theta_{x}}{\theta_{y}}$ exist? X $\sim$ exp($\theta_{x}$), Y $\sim$ exp($\theta_{y}$)

I have got the following variables. $$ X\sim exp(\theta_{x_{i}}), \ Y \sim exp(\theta_{y_{i}}) $$ and want to find the UMVUE of $$ \frac{\theta_{x}}{\theta_{y}}. $$ As the complete statistics ...
1
vote
1answer
33 views

Expected value using indicator variable

Suppose that $8$ white balls and $2$ black balls will be randomly ordered, from left to right (with all permutations of the $10$ balls equally likely), what is the expected value of the number of ...
0
votes
0answers
19 views

Question regarding conditional expectation [duplicate]

In Larry Wasseman's lecture notes(lecture 4, page 4) I found this statement $\mathbb{E}[Y|X=x] = \sum_y y f_{Y|X}(y|x)$ or $=\int_y y f_{Y|X}(y|x)dy.$ An important point about the conditional ...
0
votes
1answer
32 views

Find a copulas given joint distribution

We were given some homework to complete and would like to know how do you calculate the copulas I know from the definition that: C(X,Y)=FX,Y(Fx^-1, Fy^-1) and ill have to find the marginals by ...
0
votes
1answer
37 views

Are there situations where improper priors can be avoided via a prior on a subset of the real line and a transformation?

There are many situations where improper priors are "permissable" (Berger, 2009). In many cases, these improper priors are improper because they are "flat" on the real line. A well known example is ...
0
votes
1answer
28 views

Confidence Interval help

2 1 The contents of jars of honey may be assumed to be normally distributed. The contents, in grams, of a random sample of 8 jars were as follows: 458, 450, 457, 456, 460, 459, 458, 456 a) ...
3
votes
0answers
38 views

Algebraic Manipulations in Mann-Whitney-Wilcoxon Test Statistics

Given Let $\Delta > 0 $ be positive real number. Consider the Wilcoxon-Mann-Whitney upper tail test $H_0: \Delta \leq 0 \,\,\, \text{vs} \,\,\, H_a: \Delta > 0$ aimed at testing the difference ...
4
votes
1answer
39 views

Poisson question that I can’t solve [closed]

1.Customers arrive at a bank at a poisson rate. Suppose that two customers arrived during the first hour. What is the probability that (A) both arrived during the first $20$ minutes? (B) at least ...
7
votes
0answers
132 views

Rao-Blackwellization in variational inference

The Black box VI paper introduces Rao-Blackwellization as a method to reduce the variance of the gradient estimator using score function, in section 3.1. However I don't quite get the basic idea ...
1
vote
1answer
23 views

PCA Marginal Distribution

In PCA, if I have a latent $\vec{y}$ with loading matrix $\Lambda$, then the PCA models using: (1) $P(\vec{y}) \sim N(\vec{0}, I)$, $P(\vec{x}|\vec{y}) \sim N(\Lambda \vec{y}, \psi I)$ (2) $P(\vec{y}...
0
votes
1answer
17 views

2-sample bootstrap hypothesis test - comparing locations but different estimators in two samples

I have two independent samples X and Y where $x_i \sim F$ and $y_i \sim G$. Two different estimators A and B map X to $x_0$ and Y to $y_0$ respectively. I'd like to compare $x_0$ and $y_0$. The ...
3
votes
1answer
120 views

Why are rewards scaled when using Reinforcement Learning (RL) algorithms in practice?

I was going through this tutorial in pytorch and saw the following code: ...
2
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0answers
26 views

Covariance of a covariance matrix [closed]

Given that covariance matrix, why is the covariance of Y and Z in this case "(-1 0)" or what would be the covariance of X and Y?
2
votes
2answers
55 views

Nonparametric Sign Test

Given Likert scale survey responses: ...
-1
votes
1answer
23 views

bell curves mean,median, modes, standard deviation [closed]

Which of the bell curves in the histogram has the highest mean, median, mode & standard deviation?
0
votes
0answers
19 views

Linear discriminant analysis against quadratic discriminant analysis behavior in R

I am using linear discriminant analysis (LDA) and quadratic discriminant analysis (QDA) in R. I am working on a small data set of 4 observations and two variables <...
1
vote
1answer
46 views

Maximum A Posteriori Estimate

The formula for calculating the MAP estimate of a particular parameter, $p$, is given by the following: $p^{MAP} =$ argmax $P(p)P(p|x)$. Now I am trying to do a question where I am told the prior ...
2
votes
1answer
42 views

Expectation of Sufficient Statistic

Consider $X \sim B(n,p)$ with pmf $P(X=x) = {{n}\choose{x}} p^x (1-p)^{n-x}$. The general exponential form of an exponential family distribution is $p(x|\theta) = f(x) g(\theta) e^{\phi(\theta)^T T(...
2
votes
4answers
47 views

Question regarding independence of events

I am reading Wasserman's book and his class notes. In the notes, I found the following statement If A, B are disjoint, both having positive probability, then A and B cannot be independent. Now, if ...
0
votes
1answer
42 views

Conditional Probability - Drawing balls from an urn

I have that question from a past exam (without answer): There are two urns, say I and II. Urn I contains 1 white ball and 1 black ball. Urn II containts two white balls and 3 black balls, and suppose ...
1
vote
1answer
65 views

Gamma Distribution Sufficient Statistics

I've been asked to show the gamma distribution can be written in the form $p(x|\alpha, \beta) = f(x) g(\alpha, \beta) e^{h(\alpha,\beta)^T T(x)}$ where $T(x)$ is a sufficient statistic. .... I have ...
0
votes
0answers
31 views

Bayes Rule Bayesian Risk and Decision

Good day, When attempting this problem I came across some difficulties. A humanitarian charity wishes to classify a village as being at either high or low risk of flooding. The following ...
4
votes
0answers
33 views

Uniform distribution on the simplex. - Thomas cover

I'm trying to formulate the solution for the following problem: I was thinking in finding the equivalent distribution on $X_i$ based on $Y_i$, but I think I'm cheating. I think that the autor wants ...
0
votes
0answers
10 views

Help with centered multiple linear models

For the multiple linear model in centered form prove that where is the L-2 norm, defined as Any help would be greatly appreciated. Perhaps links to papers, chapters, websites that could help me ...
0
votes
1answer
62 views

How can I test this distribution?

Say I have 40 people that are sorted into 20 pairs. Out of the 40 people, 20 are A People and 20 are B People. I want to see if the distribution of A People and B People is random. For example, if it ...
4
votes
2answers
142 views

Different solution for a probability question

I got the following problem: Find the probability that for two arbitrary numbers $x$ and $y$ with $x,y \in [0,1]$ they satisfy $x+y<1$ and $xy<\frac1{10}$. In short words the sum of the two ...
1
vote
1answer
30 views

Random variables - proof of convergence in probability

I've got this exercise from lecture notes, but I couldn't find an answer. For each positive integer $n$, let $X_{n}$ be a non-negative random variable with $\mathbb{E}[X_{n}] < \infty$. Prove that ...
0
votes
1answer
33 views

Need help with Least Squares Estimator

I am interested in a One-Way ANOVA model:                           &...
0
votes
1answer
14 views

Summation Bounds When Finding Transformation of 2 Poisson Random Variables

I am reviewing some material on functions of several random variables from Section 7.4 of John E. Freund's Mathematical Statistics, 6th Edition, and I'm stumped on how the author gets the upper bound ...
3
votes
1answer
40 views

Calculating variance of poisson distributed random variable

I am calculating variance of a Poisson distributed random variable with mean $\lambda$. I am doing it in the following way: $\mathbb{V}(X) = \mathbb{E}(X^2) - \lambda^2 \\ = \sum_{x\geq 0} \quad x^2\...
2
votes
0answers
20 views

Trouble understanding derivation of probability for continuous time markov chain

I'm working on exercise 6.10 from "Introduction to probability models" by Sheldon M. Ross. There's an expression for the probability $P_{00}(t)$ that I don't understand. Here's the relevant ...
0
votes
0answers
28 views

Confidence Interval of shifted exponential

. I know what confidence intervals are but can a confidence interval of given size have different lengths?