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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.

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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: ...
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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 ...
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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 $$X_t=\phi X_{t-1}+\epsilon_t, ~~~~~~~~~~\epsilon_t \sim WN(0, \sigma^2)$$ ...
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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 \;\;\; \&...
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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 ...
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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(...
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16 views

Data smoothing through binning?

I have a small dataset of size n=26. ...
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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 ...
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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 ...
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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 $...
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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 ...
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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 ...
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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 ...
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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} { ...
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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$(X_i, Y_j) = 0$?
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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 ...
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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: ...
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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? ...
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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 ...
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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: ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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-...
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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? $o_p(X_n) + o_p(Y_n) = o_p(|X_n - Y_n|)$
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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-...
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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 ...
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1answer
13 views

Finding the probability of a type II error for a binomial distribution

For a binomial distribution, the hypothesis $H_0: p = 0.2$ and $H_1: p\ne0.2$ are tested at the $10%$ level. $20$ trials are performed and the critical region is $X = 0$ or $X > 7$. Calculate the ...
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25 views

Expected number of steps in Gambler's ruin game with two players

Let's say we have two players A and B. Player A has 3 coins and player B has 5 coins. If player wins the other player gives one coin. During game second player probability of loosing is $2/3$, while ...
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VaR/inverse cdf of transformation of normal variables

I have the following exercise to solve as good preparation for an exam: NOTE: $VaR_p(X)$ = Value at risk = $F^{-1}_X(p)$ Consider the bivariate normal random vector $(X_1, X_2)$. The marginals are ...
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1answer
13 views

Expected Return to the Origin – Interpreting an expectation formula

I am trying to get a head start on the next semester at uni. The following question is based on the statistical problem and solution outlined on pages 3 to 5 of this book. The problem is based on ...
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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,...
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1answer
64 views

Basic calculations with Order Statistics

I've come across the following problem, and I am tempted to delve into order statistics to solve this. I would greatly appreciate any help! Suppose you draw 6 independent samples from a continuous ...
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1answer
34 views

Self-Study: Function of a Gaussian RV

I am a beginner, solving a preparatory examen to study, and I have the following problem, where i don't understand how to start to find the answer. Is it a transformation one to one, or not? I'm ...
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0answers
13 views

Find supremum of Type II error in Neyman-Pearson framework

Let $X_1,\dots,X_n$ be an iid sample from an $N(\theta,1)$ distribution. We want to test $H_0:\:\theta=0$ against the alternative $H_1\:\theta \neq 0$ using the test statistic $$T_n(X_1,\dots,X_n) = \...
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3answers
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Calculate the variance of $\sum\limits_{i=1}^{n-1} \sum\limits_{j=i+1}^n S(X_i - X_j)$ for $X_1,\ldots,X_n$ i.i.d. random variables

In p.88 of Wand & Jones (1995), they asked to show the following result. Let $X_1,\ldots,X_n$ be a set of i.i.d. random variables and define $$U=2n^{-2}\sum_{i=1}^{n-1} \sum_{j=i+1}^n S(X_i - ...
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1answer
32 views

How to Plot the density of Z if : “Z = Gaussian RV + Discrete RV ”?

I want to do a hypothesis testing excersise that comes after, but i'm a little bit confused about the plot of the density of Z, which i feel i need to understand first: More specifically, where do i ...
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22 views

What is the probability that the drug synthesized is effective

An experimental protocol developed by NuGenCanPharm Inc to test if a cancer drug is effective is correct 99% of the time, on both effective and ineffective drugs. NuGenCanPharm Inc synthesizes 10,000 ...
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28 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 ...
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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 ...
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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....
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1answer
15 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 $...
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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. ...
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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)^...
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1answer
26 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
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1answer
43 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 ...
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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 ...