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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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0answers
6 views

Standard variance of bivariate normal distribution plus normal distribution

Problem: $W = -27 + 0.3X + 0.45Y + E$ The pair $\begin{bmatrix} X \\ Y \end{bmatrix}$ behaves like a bivariate normal with vector of averages $\begin{bmatrix} 156 \\ 86 \end{bmatrix}$ and ...
2
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0answers
21 views

Joint distribution function

Suppose that we have two indpendent random variables $X, Y$ with a joint probability density function $f(x,y)=1$, $-y<x<y$, $0<y<1$ How can I calculate the cumulative joint probability ...
1
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0answers
18 views

Probability you end up at the origin after taking 2n steps?

Starting at the origin on the line we take a step of unit to the left or to the right with probability 1/2. We do this repeatedly with independent steps. If we take 2n steps, what is the probability ...
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2answers
15 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 ...
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1answer
35 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
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1answer
24 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 ...
2
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1answer
36 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 $...
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0answers
19 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 ...
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0answers
20 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
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1answer
32 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 $c_{\alpha}, d_{\alpha}$ ...
1
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2answers
34 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(...
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1answer
25 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
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0answers
18 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
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1answer
14 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
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1answer
22 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
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2answers
74 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
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2answers
49 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....
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0answers
4 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
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1answer
16 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?
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0answers
12 views

The F Distribution Help - Did I answer the blanks correctly? [on hold]

The F Distribution Help - Did I answer the blanks correctly?
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0answers
14 views

when are probability laws for events different from those of RVs? [on hold]

I had seen rules such as $P(A\cap B) = P(A|B) P(B)$, and similar laws such as $P(a,b) = P(a|b)p(b)$. In fact, I had generally disregarded the difference between these until now, incorrectly thinking ...
2
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0answers
19 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: ...
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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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0answers
21 views

How to obtain a trend estimate using only ols?

I have data on $y$ and $t$ and I want to estimate the trend in $y$ that is supposedly given by relation $y=a(1-b e^{t/τ})$. I want to use linear regression on some transformed version of the model in ...
1
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1answer
26 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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0answers
31 views

Joint cumulative distribution of independent random variables [closed]

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 ...
1
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2answers
38 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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0answers
17 views

Data smoothing through binning?

I have a small dataset of size n=26. ...
0
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1answer
27 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
26 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
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1answer
38 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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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
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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 ...
1
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0answers
40 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} { ...
1
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2answers
44 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$(X_i, Y_j) = 0$?
1
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1answer
72 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
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1answer
31 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
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1answer
17 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
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1answer
30 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
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 ...
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0answers
26 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
26 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
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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
36 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
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1answer
86 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-...
2
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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? $o_p(X_n) + o_p(Y_n) = o_p(|X_n - Y_n|)$