Stack Exchange Network

Stack Exchange network consists of 175 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
2answers
19 views

Finding the probability

Let $E,F$ and $G$ be three events such that the events $E$ and $F$ are mutually exclusive, $P(E\cup F)=1$, $P(E\cap G)=1/4$ and $P(G)=7/12$. Then $P(F\cap G)=?$ My attempt: Since $P(E\cup F)=1$. It ...
0
votes
1answer
43 views

Variance of linear combination of Normal distributions

A company that develops software received an order for a service to be performed within a week and, in order to decide on the profile of the team of programmers to be used, it should take into account ...
0
votes
1answer
30 views

Distribution of the mean of normally distributed data

I have an exercise which requires the following: In water production, 1000 ml bottles are filled. The actual fill content is a random variable X. n = 20 bottles have been sampled (independent ...
3
votes
1answer
64 views

If all trimmed means are equal does this imply equal distributions?

I am trying to prove the following: Given that $\forall \alpha\in [0,1]$: $$\int_{F_S^{-1}(\alpha)}^{\infty}xf_S(x)\,dx = \int_{F_0^{-1}(\alpha)}^{\infty}yf_0(y)\,dy$$ where $F_S^{-1}(\alpha)$ and $...
0
votes
0answers
24 views

Bootstrap confidence interval for just-identified IV estimator

Assume we have a regression model \begin{equation} y_{i} = z_{i}' \delta + \epsilon_{i} \end{equation} with dependent variable $y_{i}$, L regressors $z_{i}$ and K instruments $x_{i}$, and assumptions ...
1
vote
1answer
39 views

Conditional probability questions

Three dice are rolled. If no two show the same face, what is the probability that one is an ace? Given that a throw with ten dice produced at least one ace, what is the probability p of two or more ...
2
votes
1answer
68 views

UMAU confidence interval for $\theta$ in a shifted exponential distribution

Suppose $X_1,X_2,\ldots,X_n$ is a random sample drawn from the distribution $$f_{\theta}(x)=e^{-(x-\theta)}\mathbf1_{x>\theta}$$ It can be shown that there exist some $c_{\alpha}, d_{\alpha}$ ...
2
votes
2answers
68 views

Conditional Probability and Expectation for Poisson Process

To solve part (a) I have $P(X_2 = k\mid X_1 = 1)= \dfrac{P(X_2 = k \cap X_1 = 1)}{P(X_1 = 1)} = \dfrac{e^{-2}}{e^{-1}}=e^{-1}$. Then for part (b), for simplicity, I let $X_2=X$ and $X_1=Y$, then $$E(...
0
votes
1answer
32 views

A question about ANOVA

Question: My problem: Regarding the above question, a solution set was provided(although it was not explained by anyone). There the problem was solved by "One way ANOVA" model, Yij=u+eij, where ...
0
votes
0answers
21 views

Likelihood is “proportional to a probability”. Which one? [duplicate]

In various places (see quotes below) it says that the likelihood is "proportional to a probablility". Which probability is it proportional to? In the context of Bayes theorem, it is not proportional ...
1
vote
1answer
31 views

Probability that one random variable using the Beta Distribution being greater than another, bounded intervals

I am doing some practice problems to prepare for my statistics exam, and I just want to know if my reasoning is correct on one problem, and if not, I want to know how I should reason through this. The ...
1
vote
1answer
30 views

Where is my potential flaw in my Z-test of Proportions?

I was working through this z-test of proportions example I found online. The online example solutions says that the difference between the groups is statistically significant, whereas I concluded it ...
1
vote
2answers
88 views

Likelihood is not “proportional to” a single probability density?

In various places it says that the likelihood (e.g. in the Bayes formula) is "proportional to a probablility". For example https://alexanderetz.com/2015/04/15/understanding-bayes-a-look-at-the-...
3
votes
2answers
50 views

Having difficulty deciding limits of integration for a joint to marginal pdf

A joint pdf, $f_{X,Y}(x,y)=5$, is given with the following intervals: $-1<x<1$ $x^2<y<x^2+{1\over{10}}$ I am trying to find marginal pdf of $f_Y(y)$ but I am stuck. Trying for hours....
0
votes
0answers
8 views

Econometric Models in Economic Development Theory

I have been reading some Economic Development and Policy Research papers and I realise there is extensive employment of econometric tools and models. For example, one of the papers used an Oaxaca ...
0
votes
1answer
18 views

Meaning of “card”

The following statement is made in Elements of Statistical Learning: "Our loss function can be represented by a K × K matrix L, where K = card(G)." What does card(G) mean?
2
votes
0answers
27 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
1answer
49 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 $$X_t=\phi X_{t-1}+\epsilon_t, ~~~~~~~~~~\epsilon_t \sim WN(0, \sigma^2)$$ ...
2
votes
1answer
72 views

Joint cumulative distribution of independent random variables

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 \;\;\; \&...
0
votes
1answer
49 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
55 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(...
0
votes
1answer
50 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
28 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 ...
1
vote
1answer
46 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
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 ...
1
vote
0answers
53 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 ...
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} { ...
4
votes
2answers
77 views

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

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$(X_i, Y_j) = 0$?
1
vote
1answer
140 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 ...
0
votes
1answer
39 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: ...
2
votes
1answer
22 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? ...
1
vote
1answer
49 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: ...
0
votes
1answer
28 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
31 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 ...
1
vote
0answers
51 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 ...
1
vote
0answers
25 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 ...
0
votes
0answers
37 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 ...
3
votes
1answer
93 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
0answers
29 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
46 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
26 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
36 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 ...
0
votes
1answer
29 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 ...
1
vote
0answers
49 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 ...
2
votes
0answers
26 views

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 ...
0
votes
1answer
26 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 ...
1
vote
1answer
68 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,...
5
votes
1answer
70 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 ...
-1
votes
1answer
35 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 ...
0
votes
0answers
14 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) = \...