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.

1
vote
0answers
11 views

Is my interpretation correct for these residuals plots?

In preparation for my exam, I'm trying to interpret the residuals in order to understand if the time series has been modelled correctly. Otherwise, I have to suggest an improvement. Here is the text: ...
1
vote
0answers
32 views

What's the variance of an AR(1)/ARCH(1)

The main question is: an AR(1)/ARCH(1) process has the variance of the ARCH(1)? I've tried to compute the unconditional variance of an AR(1)/ARCH(1) model, so an AR(1) in which the noise is modelled ...
-1
votes
0answers
28 views

Joint cumulative distribution of independent random variables [on hold]

X,Y,Z are non negative random variables which are independent and uniformly distributed in [0,1] and let $\alpha$ be a given number in [0.1]. Now how to compute $\text{Pr}(X+Y+Z>\alpha \;\;\; \&...
15
votes
1answer
10k views

A proof for the stationarity of an AR(2)

Consider a mean-centred AR(2) process $$X_t=\phi_1X_{t-1}+\phi_2X_{t-2}+\epsilon_t$$ where $\epsilon_t$ is the standard white noise process. Just for sake of simplicity let me call $\phi_1=b$ and $\...
3
votes
0answers
59 views

Posterior mean estimator with MCMC (Metropolis Hastings Algorithm) - Concrete example

I have a little project for which I have to estimate parameters on a PSF (Point Spread Function = response of the system to a dirac, i.e a star in my case). I have the 6 parameters to estimate : $p=(\...
1
vote
1answer
23 views

Expectation, variance and autocorrelation of a “complex” AR(1) function

I'm preparing the exam for "stochastic models" and I encountered this exercise which is giving me a lot of problems: Let $$X_t=\phi X_{t-1}+\epsilon_t, ~~~~~~~~~~\epsilon_t \sim WN(0, \sigma^2)$$ ...
1
vote
1answer
29 views

doubtful regarding my solution to bonferroni's principle exercise

Self learning and not quite good at probablility and statistics, my question is regarding solution to exercise 1.2.1 in chapter 1 of Mining of Massive Datasets book. The text of the exercise reads: ...
-1
votes
0answers
14 views

Would like a double check on expected value & variance problem

If a student randomly chooses the answer to two multiple choice questions, where the first question has 3 possible answers and the second has 5, find: 1) $E(X)$ 2) $Var(X)$ For 1, I believe the ...
3
votes
1answer
83 views

Proving that given Markov chain is homogeneous. Find state space and transition matrix

Let $X_i$ be the results of a consecutive throws of a die. Let $Z_n=3(X_1^2+\cdots+X_n^2) \bmod 5$. Show that the sequence ${\{Z_n \mid n\geq1\}}$ is a homogeneous Markov Chain. Find a state space and ...
0
votes
1answer
44 views

Proof of probabilities that may not be independent

I am given the problem: Given $P(A) = \frac{3}{4} $, $P(B) = \frac{3}{8} $, show that: a) $P(A or B) > \frac{3}{4} $. b) $\frac{1}{8} < P(A and B) < \frac{3}{8} $. The problem does not ...
1
vote
2answers
35 views

Maximum Likelihood Estimator (MLE) for $2 \theta^2 x^{-3}$

I'm having a bit of trouble solving this. $$ f(x_i; \theta) = 2 \theta^2 x_i^{-3}, 0 \le \theta \le x_i \lt \infty $$ I start by finding $f(\textbf{x}; \theta)$: $$ f(\textbf{x}; \theta) = \prod{f(...
24
votes
6answers
11k views

What's the difference between logistic regression and perceptron?

I'm going through Andrew Ng's lecture notes on Machine Learning. The notes introduce us to logistic regression and then to perceptron. While describing Perceptron, the notes say that we just change ...
3
votes
1answer
336 views

What is the best statistical test for skewed distributions with different spread? (See image)

I am working on an exercise in which I have to study how does one specific feature of a listing on a website for house rentals affects the number of reservations. I plotted my kernel density ...
-1
votes
0answers
16 views
1
vote
1answer
32 views

Covariance of Random Proportions in Multinomial Counts

In Agresti's Categorical Data Analysis Second Edition, at Section 14.1.4, there is a proof of the Asymptotic Normality of Functions of Multinomial Counts. It is stated that for a vector of responses $...
0
votes
1answer
26 views

Mean of an ARMA(1,1) model

Let $X_t$ be a weak stationary process ARMA(1,1) $X_t=c+\phi X_{\left(t-1\right)}+\theta\varepsilon_{\left(t-1\right)}+\varepsilon_t$ $\varepsilon_t$ ~ $WN\left(0,\sigma^2\right)$ The estimated ...
0
votes
0answers
25 views

Independent Study Statistics/Probability Grad Level [duplicate]

I am trying to decide on topics for my independent study this semester. I am a Pre-Doctoral Mathematics student, so looking for a more math based text rather than engineering based (which I have found ...
2
votes
1answer
305 views

Conway–Maxwell–Poisson (CMP) distribution and exponential family

So I have a question here about the CMP distribution: My understanding is that $b(\theta)$ is only a function of $\theta$ but why is $v$ able to be included in that function, would $v$ not be a ...
3
votes
1answer
86 views

Can an independent t-test be used on paired data when the pairing is unknown?

Suppose the effectiveness of a training course is examined, and performance of each individual in a group is taken both before and after, and the differences are compared in a paired $t$-test. Would ...
0
votes
0answers
20 views

Strange error in fitting classifier [migrated]

I'm working through O'Reilly's Hands-On Machine Learning with Scikit-Learn & Tensorflow. I'm working on training a classifier on the MNIST dataset and I'm getting the error ...
0
votes
1answer
22 views

Expectation conditional on self and others

I would simply like to know if: $E[x_1|x_1,x_2]=E[x_1|x_2]$ or $E[x_1|x_1,x_2]=E[x_1|x_1]=x_1$ or something completely different and why. This is not homework. It came up because I'm trying to ...
3
votes
1answer
625 views

Hoeffding's Inequality

I am studying the feasibility of learning from the book Learning from Data. The author uses a bin analogy to discuss the feasibility of learning in a probabilistic sense. I have certain questions to ...
0
votes
0answers
25 views

States of Markov chain and stationary distribution

Let $X$ be a Markov chain with a state space $S={\{0,1,2,... \}}$ and a transition matrix $P$ with given $p_{i,0}=\frac{i}{i+1}$ and $p_{i,i+1}=\frac{1}{i+1}$, for $i=0,1,2,...$. Find out which states ...
0
votes
1answer
340 views

How to minimize $Variance(S)$?

Suppose $X_1,....,X_n$ are $iid$ random variables and for each of them $Variance(X_i)= \sigma^2$. $a_1...a_n$ are also real numbers and $\sum_{i=1}^n a_i = 1$ If $S = \sum_{i=1}^na_iX_i$, prove $...
7
votes
2answers
597 views

Minimum Training size for simple neural net

There's an old rule of thumb for multivariate statistics that recommends a minimum of 10 cases for each independent variable. But that's often where there is one parameter to fit for each variable. ...
0
votes
1answer
3k views

Effects of different changes on test statistic

Will each of the following increase, decrease, or have no effect on the value of the test statistic in a one-independent sample t-test? The sample size is doubled The level of significance is ...
1
vote
2answers
313 views

What is the empirical frequency?

I have generated $1.000.000$ repetitions of the experiment of drawing $20$ iid. Bernoulli random variables $X_1, ..., X_{20}$ (20 coins) with bias $1/2$ in R. I then wish to plot the empirical ...
0
votes
0answers
13 views

Joint distribution of a two part model

Let $ Y $ be a random variable defined on $ (0, +\infty) $. In a univariate two part model, the distribution of $ Y $ is defined as follows \begin{equation*} g ( y_i ) = \left\{ \begin{array} { ...
1
vote
1answer
68 views

Probability Density function of Poisson distribution

This is an assignment I got for my course on Stochastic Processes: Let us consider a random variable X distributed as a Poisson P (λ) where λ ∼ [0.5, 1]. (a) Which are the unconditional ...
1
vote
2answers
43 views

Suppose $\mathbf{X, Y}$ are independent random vectors. Are their components independent?

Let $\mathbf{X} = (X_1, \dots, X_p)^\top$ and $\mathbf{Y} = (Y_1, \dots, Y_p)^\top$ be independent. Does it then follow that $X_i$ is independent with $Y_j$ i.e. cov$(X_i, Y_j) = 0$?
0
votes
1answer
342 views

Deriving Transition Matrix of the Embedded Markov Chain given the generator matrix?

Full Problem: A continuous-time Markov chain has generator matrix $$Q= \begin{pmatrix} -1 & 1 & 0 \\ 1 & -2 & 1 \\ 2 & 2 & -4 \\ \end{pmatrix} $$ (i) ...
2
votes
2answers
308 views

Specification of logical node (with distribution?) in WINBUGS

For a piece of homework I have an assignment using WINBUGS which I must admit confuses me to say the least. Tangential to my question but I have a few stochastic nodes that are to be gamma ...
2
votes
3answers
9k views

Are Event and Outcome synonymous?

Outcome : An outcome is a result of a random experiment. Event : A single result of an experiment. Are Event and Outcome synonymous ?
2
votes
1answer
16 views

Derivation of the Mann Whitney U normal approximation

The normal approximations for the Mann Whitney U statistic are given by wikipedia but there are no refrences mentioned. What are the actual derivation steps of the untied and tied case approximations? ...
0
votes
1answer
85 views

Variational autoencoder obtain high likelihood but produce low quality sample?

I'm watching Ian Goodfellow's introduction to generative models. When he was introducing variational autoencoders at 22:29, he said: Variational autoencoders are good at obtaining high likelihood,...
3
votes
1answer
45 views

How to choose an appropriate variational distribution?

I work in deep learning research and I am trying to learn how to use variational inference in order to approximate a posterior over the learned weights. I have looked extensively at Yarin Gal's ...
0
votes
1answer
28 views

Compute $P(2\leq x\leq 8)$ with Poisson distribution and $\lambda=7.2$

Compute $P(2\leq x\leq 8)$ with Poisson distribution and $\lambda=7.2$ My attempt: I need calculate this using $R$. then I use this: ...
0
votes
0answers
17 views

Logistic regression model interpretation-Adult data set [closed]

I am very new in the field, with no idea about statistics or RStudio. However, I have to work on an assignment in R on the Adult data set: "1) Create a model to predict whether someone has an income ...
0
votes
1answer
27 views

Probability of finding a lost item

I am trying to solve the following problem and was wondering if someone can verify my answers. Big Joe has lost an important document. There is a 70% probability it is at home, and a 30% chance it is ...
0
votes
0answers
25 views

Derive the estimator for the integrated squared bias $\int \left(\operatorname{E}\hat{f} - f\right)^2 $

This problem is found in p. 77 of Wand & Jones' (1995) book. If you are familiar with nonparametric estimation you may skip this introduction. Suppose we want to minimize the integrated squared ...
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}...
1
vote
1answer
61 views

Most powerful test for deciding probability mass function

Let $X$ be an integer valued random variable supported on $\{0.1.2...,12\}$ whose pmf is either $g(x)=1/13; x=0,1,...,12$ or $ f(x)=\dfrac x {36} 1_{\{0,1,...,6\}} + (\dfrac 13 - \dfrac x{36})1_{\{7,...
1
vote
0answers
23 views

Solution verification for a hypothesis testing question

I am posting this question as a solution checking. Let $X_1,...,X_{30}$ be a random sample from the exponential distribution with unknown mean $\mu\in \{1,1/\delta\}$ (where $\delta>1)$. Consider ...
6
votes
1answer
3k views

Degenerate random variable

Let $X$ and $Y$ be independent $rv$ such that $XY$ is a degenerate $rv$. Can I say that individually $X$ and $Y$ are also degenerate? Why?
0
votes
0answers
35 views

Which one of these is correct for linear regression?

Only one of these is supposed to be the correct one for simple linear regression. Which pair of plots would you say has constant variance and normal distribution? I feel like none of them have both ...
0
votes
0answers
23 views

How do you express ARIMA(2,1,2) in terms of the backshift operator?

I've so far achieved the following: $$y_t-y_{t-1}=\phi_1y_{t-1}+\phi_2y_{t-2}- \theta_1e_{t-1}-\theta_2e_{t-2}+e_t$$ Therefore Yt-BYt=(phi)Byt+phiB^2yt-B(theta)et-B^2(theta)et+et Yt-BYt-(phi)Byt-...
2
votes
2answers
41 views

Rate of convergence of sum of two random variables

Let $X_n$ and $Y_n$ be random variables such that $X_n=o_p(1)$, $Y_n=o_p(1)$, $X_n - Y_n = o_p(1)$. Is the following correct? $o_p(X_n) + o_p(Y_n) = o_p(|X_n - Y_n|)$
0
votes
0answers
17 views

When we are proving why ARIMA(0,1,1) is equal to simple exponential smoothing, why can we considered theta to be equal to (1-alpha)

I know this is a very basic question, but its not clarified within my lectures. Essentially when you have ARIMA(0,1,1) You can simplify the theta 1 term in order to obtain SES via stating its (1-...
3
votes
1answer
345 views

Unbiased estimate for test error in Leave-One-Out Cross validation

I know there are many related questions to this but none are exactly to the point I want to ask , My question is in terms of simple linear regression, This statement(In bold)from the book An ...
3
votes
1answer
33 views

What is the probability that at least three guilty parties are caught at the same time and at least four of the innocent are released?

A lie detector will be used by police to investigate 10 suspects of involvement in a particular crime. Admit that among them, five are guilty (but will plead innocence) and the other five are really ...