Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
0answers
29 views

How do we compute OLS coefficients from Sum of Squares in multiple linear regression?

Suppose we have a variable y which is dependent on 2 variables say x1 and x2. Then I can understand how we can compute Sum of squares due to x1 and x2. For instance we may have Type I Sum of squares ...
1
vote
0answers
45 views

Plot the exact density of a transformation of a distribution

I would like to compute (and plot) the exact density of the following distribution: $ X_i \sim exp(-Exponential(\lambda)) - 0.5 $ I already have the estimated density for this distribution, but I ...
2
votes
1answer
43 views

Newsvendor problem with Poisson demand

Problem: The demand for a particular weekly magazine in a newsstand follows a Poisson distribution averaging 5.2 copies per week. The value paid for each magazine is 15.00 and the sale price is 30.00....
2
votes
1answer
16 views

Covariance of An Empirical Distribution Function Evaluated at Different Points

The problem is extracted from All of Statistics (Exercise 7.5), Larry Wasserman. I don't have a solution manual to the book so I post here the problem together with my attempted answer: Let $x$ and $...
0
votes
0answers
16 views

Conflicting “facts” about the likelihood employed in Bayes theorem [duplicate]

Consider the following "facts" about Bayes theorem and likelihood: Bayes theorem, written generically as $P(A|B) = \frac{ P(B|A) P(A) }{ P(B) }$ involves conditional and marginal probabilities. ...
0
votes
0answers
38 views

Understanding sum of square deviations [duplicate]

Given $X_1...X_n\stackrel{iid}{\sim} N(\mu,\sigma^2)$ and $U=\sum_{i=1}^n (X_i-\overline{X})^2$, why is $U\sim\sigma^2 \chi_{n-1}^2$ ? And what would be the distribution of $V=\sum_{i=1}^n (X_i-\mu)^...
0
votes
1answer
27 views

Is a Kalman filter ever the optimal way to estimate a dynamic value given a full history of measurements?

I'm trying to get some intuition for Kalman filtering, and I conceived this toy example: Say that I have a sensor that tracks a moving 1-dimensional target. Say that the measurements from the sensor ...
2
votes
1answer
44 views

Is the likelihood in Bayes theorem a probability? [duplicate]

First some notation and definitions: In the Bayes formula as written for machine learning applications, $$ p(\theta|D) = \frac{ p(D|\theta) p(\theta) }{ p(D) } $$ commonly $p(\theta)$ is labeled ...
0
votes
0answers
84 views

EM Algorithm for Bayesian Networks with missing data

Setting: learning parameters of Bayesian Network (BN) with missing data. Algorithm: Expectation-Maximization. Question: suppose I am in the M-step, and that in the complete data there are no ...
0
votes
0answers
33 views

Likelihood of Gamma Distribution

I'm studying bayesian stats on my own and came across the following problem on Coursera. Can anyone help me understand how to work through this? $x_{i}\stackrel {iid}{\sim} Beta(\alpha,\beta), i=1.......
1
vote
1answer
56 views

Standard Error of a function of ML estimators

The background of the problem is as follows: Suppose $X_1,...,X_n \sim U(a,b)$ independently where $a$ and $b$ are unknown parameters and $a < b$. Let $\hat\tau$ be the MLE of $\tau$, where $\tau =...
0
votes
0answers
31 views

How to determine the line which a time-series fluctuating around

Let $X_t=1+3t+0.5X_{t-1}+ \epsilon_t$ be a trend-stationary model, where $\epsilon$ is a white noise, which has zero expected value and standard deviation. Which line is the time series fluctuating ...
1
vote
1answer
42 views

Residual Sum of Squares degrees of freedom intuition [duplicate]

Let RSS = Residual sum of squares $ = \sum (y_i - \hat{y}_i)^2$. Without proof, $\frac{RSS}{\sigma^2} \sim \chi^2_{n-2}$. I do not quite understand why the DoF is $n-2.$ Could someone explain?
1
vote
0answers
26 views

What is the probability that the new commitment will be fulfilled?

A consulting firm was hired to develop an Engineering project. Based on their previous experience, the direction of this office knows that the time (in months) needed to perform this type of task ...
1
vote
0answers
49 views

Finding Chernoff bounds maximum estimators

I am currently trying to resolve the following exercise about Chernoff bounds: Let $X_{1}, X_{2}, \dots, X_{n}$ be independent, identically distributed (i.i.d) random variables with distribution $N(0,...
2
votes
1answer
82 views

Expectation of reciprocal of $(1-r^{2})$

I tried finding expectation from the density function but then realised that I was solving with the density function of $r$ and not it's square. I don't know the density function of $r^{2}$. I am ...
1
vote
1answer
26 views

How to show that $\frac{1}{\theta}$ is a flat prior for $\log\theta$? [duplicate]

I do not really understand what the statement even means. To my understanding, a prior $p(\theta)$ is said to be flat if $p(\theta) =$ constant $\forall \theta,$ where $p(\theta)$ is the prior ...
0
votes
1answer
50 views

Canonical link of Gamma Distribution [duplicate]

I wonder why my professor said that Gamma's canonical link is $\frac{1}{\mu}$. My thoughts are: EDIT: $\theta$ is the canonical parameter. Since $$\mathbb{E}_\theta(Y)=b^{'}(\theta)=-\frac{1}{\theta}...
0
votes
0answers
17 views

MLE Asymptotic Theorem [duplicate]

Statement: if $\hat{\theta}$ is the MLE of some parameter $\theta$ then asymptotically $\hat{\theta} \sim \mathcal{N}(\theta, I_\theta^{-1}),$ where $I_\theta$ is the fisher information matrix. My ...
0
votes
1answer
44 views

Standard error of $\widehat\theta_1 + \widehat\theta_2$

I am working on some sample questions, and I came across one I have no clue how to answer. How do Standard error of $\widehat\theta_1$, $\widehat\theta_2$, $\widehat\theta_1 + \widehat\theta_2$, ...
2
votes
1answer
60 views

CDF and MGF of a Sum of a discrete and continuous random variable

I am currently dealing with the following exercise: Given the random variables $X \sim Be(p), Y \sim Exp(\lambda)$, and assume they are independent. Set $Z:= X + Y$. Compute the Moment Generating ...
0
votes
0answers
17 views

Explaining Low Predictive Ability

For a university project I am supposed to predict one survey question (with 5 different answers) using 20 other survey questions (each with 4 or 5 different answers i.e. all predictor variables are ...
5
votes
2answers
110 views

How to justify that $(Y_1,Y_2)$ is not bivariate normal without finding its exact distribution?

Suppose $X_1$ and $X_2$ are independent $N(0,1)$ variables. Define $$Y_1=X_1\,\text{sign}(X_2)\quad,\quad Y_2=X_2\,\text{sign}(X_1)$$ I have to show that $(Y_1,Y_2)$ is not bivariate normal ...
2
votes
1answer
64 views

Math questions in Kalman filter equation derivation

I am interested in data analysis. While my working data (actually it's shopping mall's daily sale) is accumlating, I wish to find some statistical laws underlying business phenomena. I left school for ...
0
votes
0answers
24 views

Help on concentration of measure problem

I am trying to solve the following exercise (exercise 8.1 from Bucheron, Lugosi, Massart): Use Marton's transportation inequality to show that if $P$ is a product measure on $\mathcal{X}^n$, then ...
4
votes
3answers
47 views

Likelihood function when only $\max_{1\le i\le N}X_i$ is observed and $N$ is parameter

Let $X_1,X_2,\ldots,X_N$ be i.i.d random variables having $\text{Exp}(1)$ distribution where $N$ is unknown. Suppose only $T=\max\{X_1,X_2,\ldots,X_N\}$ is observed. I have to derive a most ...
1
vote
1answer
57 views

Poisson and conditional probability

Admit that the number of participants who intend to enroll in a given training follows a Poisson distribution with a mean of $12.$ If there is not a minimum of five enrollments, training is not ...
0
votes
1answer
54 views

Computing the probability density function

Suppose we have the cdf $$F_X(x) = \begin{cases} 0 \quad \quad, x<-1 \\ 0.25 \quad \quad, -1\leq x < 1 \\ 0.5 \quad \quad, 1 \leq x < 2 \\ \frac{2}{3} \quad \quad, 2 \leq x < 3 \\ 1 \quad ...
0
votes
0answers
15 views

How do I estimate parameters of Weibull distribution in R based on a given dataset? [duplicate]

Here I got the Weibull distribution function like this, and how can I estimate the parameters by the below data set?
0
votes
0answers
19 views

Step-wise Multiple Regression or ANCOVA

I have an assignment that gives a dataset and a choice of 3 tests: Step-wise Multiple Regression, ANCOVA and Log-Linear Analysis. The dataset consists of ...
0
votes
1answer
17 views

Compound risk poisson models

I was just working through this question. A compound Poisson risk model is used to model the total claims S experienced by an insurance company over one year, of the form: $S = X_1 + ... + X_n$ ...
8
votes
1answer
164 views

Reproduce figure of “Computer Age Statistical Inference” from Efron and Hastie

The summarized version of my question (26th December 2018) I am trying to reproduce Figure 2.2 from Computer Age Statistical Inference by Efron and Hastie, but for some reason that I'm not able to ...
2
votes
0answers
28 views

Simulation - problem of maximization inside a circle

I am doing some projects related to statistics simulation using R based on "Introduction to Scientific Programming and Simulation Using R". In the Students projects session (chapter 24), I am doing ...
2
votes
2answers
52 views

Knowing correlation, bivariate normality, means and standard deviations, find values probability

After months of study I still do not get it. I apologize (see: Estimating values knowing their Pearson's r and their means and standard deviations) Imagine, for example, I have two bivariate ...
0
votes
0answers
33 views

Covariance non-zero mean AR(1)

Why when I compute the autocovariance function of a non-zero mean AR(1), X(t)-u=Φ(X(t-1)-u)+ε the presence of the mean does not change my result and so the formula should be the same of a zero-mean ...
1
vote
2answers
56 views

Confusion about range of integration for density function

Consider the joint density function: $$f(x,y) = \begin{cases} 2 & & \text{for } 0 \leq x \leq1 \text{ and } 0 \leq y \leq 1-x, \\[6pt] 0 & & \text{otherwise}. \end{cases}$$ From ...
1
vote
1answer
81 views

Convergence in probability of $\frac{1}{n}\sum_{i=1}^n X_i^2$ when $X_i$'s are i.i.d $N(0,1)$

Question: My approach: And after this I am stuck..How do I put the modulus over here and how do I determine the appropriate value of "k" ? (here k signifies the value of convergence in probability ...
2
votes
2answers
106 views

Find $P(A^2 < B)$ where $A$ and $B$ are independent and uniformly distributed $\mathrm{Unif}(0,h)$, $h > 0$

I solved it two ways and in both the cases the answer is different and different from the actual answer. Approach 1: Since, $A$ and $B$ are independent, we can find the joint distribution of $AB$ ...
0
votes
0answers
13 views

Probit Hypothesis Test: $H_0: \beta_1 \geq 1$

Problem Given the output of a probit model and no knowledge of the sample data: ...
0
votes
1answer
34 views

Intensity function in Poisson random effect model

I have a somewhat general question about intensity functions in Poisson random effect models. Consider the Poisson random effects model in which conditional on a random effect $u$, an individual ...
1
vote
1answer
58 views

Bayesian posterior for Geometric Distribution

I have the following homework problem I am trying to solve for but am stuck with the posterior part. Note the the geometric distribution is a discrete distribution that has a probability mass function ...
0
votes
0answers
9 views

two-parameter Pareto distribution with known A [duplicate]

I am trying to solve the following problem. Any help would be great: Scores are distributed as a two-parameter Pareto distribution with a=3 Scores for 3 groups are as follows: Group A has expected ...
1
vote
0answers
49 views

Critical region of distribution $f(x;\theta) = \theta x ^{\theta -1}$

I have been asked to find the critical region of the distribution given by $f(x;\theta) = \theta x ^{\theta -1}$ under the hypotheses $H_0: \theta = \theta_0 $ and $H_1:\theta < \theta_0.$ Show ...
0
votes
0answers
30 views

When calculating sum of observation in Bayes Theorem, why do we remove one outcome?

relatively quick question regarding this course question: Hearing about your brilliant success in working with M&Ms, Mars Inc. transfers you over to the Skittles department. They recently have ...
3
votes
0answers
61 views

Finding the UMVUE of $\theta^2$ where $f_X(x\mid\theta) =\frac{x}{\theta^2}e^{-x/\theta}I_{(0,\infty)}(x)$

Let $X_1, X_2, . . . , X_n$ be iid random variables having pdf $$f_X(x\mid\theta) =\frac{x}{\theta^2}e^{-x/\theta}I_{(0,\infty)}(x)$$ where $\theta >0$. Give the UMVUE of ${\theta^2}$ I ...
2
votes
1answer
51 views

Bayes estimator with weighted Loss

I have been working through a wide variety of problems involving Bayes risk and loss functions and I couldn't immediately solve the following From "The Bayesian Choice", Consider $x \sim N(\...
1
vote
0answers
65 views

Calculate a confidence interval of 88% for the difference in proportions [duplicate]

A survey of 400 men and 400 women in a certain city showed that 270 men and 240 women favor a certain proposition. Calculate a confidence interval of 88% for the difference in proportions between men ...
2
votes
1answer
136 views

Some questions about exponential families

Regarding the book The Bayesian Choice I understand most of chapter three on exponential families, but there are two parts I have trouble understanding. The first is Consider$$f(x|\theta)=h(x)\...
0
votes
0answers
27 views

Calculate $R^2$, $R^2_{adj}$, and F-statistic from $\text{R}$ model summary

I am given the full model, $M_{\tt f}$, with the regression line $$ {\tt response} = \beta_0 + \beta_1{\tt A} + \beta_2{\tt B} + \beta_2{\tt C} + \beta_4{\tt D} + \beta_5{\tt E} + \beta_6{\tt F} + ...
1
vote
0answers
50 views

Cumulative distribution function of a squared laplace random variable

I am trying to calculate $F_Y(x)$ (CDF) of $Y=X^2$ where $X$ is a random variable of Laplace Distribution $f_X(x) = \frac{1}{2}e^{-|x|}$ (let's take a simple case when parameters $\mu=0$ and $b=1$). ...