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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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How to solve an optimization problem with variable in indicator function?

How to solve the following optimization problem? $$ \underset{D, U \in \mathbb{R}}{\min} (1+a)\text{E}_{X}[(X-D)\cdot\mathbb{1}_{\{D<X\leq U\}}] - (M-D)\cdot\mathbb{1}_{\{D<M\leq U\}}, $$ where ...
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2answers
143 views

question about MSE mean square error

The following is taken literally from Wikipedia's mean squared error in the mean subheading: "Suppose we have a random sample of size $n$ from a population, $X_1$, ... ,$X_n$. Suppose the sample ...
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1answer
18 views

Estimating the effect of time-invariant and time-varying regressors using Fixed Effects

I have a question about the Fixed vs Random effects modeling. It is said that Fixed effects modeling identifies only parameters for time-varying regressors, not for time-invariant regressors I ...
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0answers
39 views

Posterior mean computation of “Monty Hall Poblem”

Background As I understand that the "Monty Hall Problem" is well studied, e.g., here or here, etc. [I am relatively new to probability theory. So, please help me to learn something from you experts ...
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3answers
185 views

If $X_1,\cdots,X_n \sim \mathcal{N}(\mu, 1)$ are IID, then compute $\mathbb{E}\left( X_1 \mid T \right)$, where $T = \sum_i X_i$

Question If $X_1,\cdots,X_n \sim \mathcal{N}(\mu, 1)$ are IID, then compute $\mathbb{E}\left( X_1 \mid T \right)$, where $T = \sum_i X_i$. Attempt: Please check if the below is correct. Let ...
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2answers
135 views

Find Bayes Estimator when Kernel of posterior is not clear

Suppose $x\mid\theta \sim \operatorname{Gamma}(\frac{n}{2},2\theta)$ and $\theta \sim$ inverse Gamma$(\alpha, \beta)$ with loss function $L(\theta, d)=\frac{(\theta-d)^2}{\theta^2}$ We wish to find ...
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1answer
133 views

Conditional Expectation of pdf

Wish to identify what I'm doing wrong when finding the $\operatorname E(X\mid Y=5)$ of the following: $$f(x, y)=\begin{cases} 1/6 & \text{if } 0<x<2, 0<y<6-3x \\ 0 & \text{...
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0answers
41 views

Proving that $V(\hat{y}_{x_0}) = \sigma^2\bigg[\frac{1}{n}+\frac{(x_0-\bar{x})^2}{S_{xx}}\bigg]$ [duplicate]

Exercise : Prove that the variance of $\hat{y}_{x_0} = \hat{b_0} + \hat{b_1}x_0$ is : $$\text{Var}(\hat{y}_{x_0}) = \frac{\sigma^2\sum x_i^2}{n\sum(x_i-\bar{x})^2}+\frac{\sigma^2x_0^2}{\sum(x_i-\...
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0answers
51 views

How to learn basics of statistics using Python?

What kind of learning material would you suggest as I would like to learn doing statistics on Python. I mean, I studied mathematics and programming but I would like to improve my statistical skills as ...
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1answer
19 views

Confusion on the computation of Leave One Out cross validation?

1) I was studying about cross-validation and have a bit of confusion here. I understand about the k-fold technique, where if you ...
1
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1answer
54 views

How big should be a sample size?

Question: Assuming the population standard deviation σ = 3, how large should a sample be to estimate the population mean µ with a margin of error not exceeding 0.5? My answer: SE<= 0.5=3/√n ---> ...
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0answers
37 views

What is the difference between Pearson correlation and dynamic correlation (DCC-GARCH)?

I often read dynamic correlation, which I believe related to DCC-GARCH. How is the dynamic correlation different comparing to the Pearson correlation?
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2answers
82 views

Showing that $\sum_{i=1}^n (y_i-\hat{y_i})(\hat{y_i} - \bar{y}) = 0$ for the generalized linear model [closed]

Exercise : Prove that for the generalized linear model, it is : $$\sum_{i=1}^n (y_i-\hat{y_i})(\hat{y_i} - \bar{y}) = 0$$ Question : How would one proceed with proving that for the generalized ...
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0answers
43 views

Is $\sqrt{y} = \beta_0+\beta_1x_1+\epsilon$ the same as $y=\beta_0+\beta_1(x_1)^2+\epsilon$?

Is $\sqrt{y} = \beta_0+\beta_1x_1+\epsilon$ the same as $y=\beta_0+\beta_1(x_1)^2+\epsilon$ ? If I am looking for the estimated coefficient for $(x_1)^2$ both equations, what should they be? Should I ...
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1answer
42 views

Bayesian statistics: probability of next point

I am reading the Deep Learning book and having some difficulties with the following formula (page 134): $$ p(X^{m+1} | x^1, \dots, x^m) = \int p(X^{m+1} | \theta) p(\theta | x^1, \dots, x^m) d\theta. ...
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2answers
74 views

Testing the general linear hypothesis: $H_0: \beta_1 = \beta_2 = \beta_3 = \beta_4 = \beta$

Again, we are testing the linear hypothesis; $H_0: \beta_1 = \beta_2 = \beta_3 = \beta_4 = \beta$ for the model, $$y = \beta_0 + \beta_1x_1+\beta_2x_2+\beta_3x_3+\beta_4x_4+\epsilon$$ I know how ...
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1answer
36 views

Transformations to Linearize a Model in Multiple Linear Regression. Deciding if a model is linear, intrinsically linear, or non-linear

I'm asked to indicate whether a model is linear. If not, I need to find a suitable transformation. Well, the model seems strange to me in that I can't imagine where a situation like this would arise. ...
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1answer
70 views

Standard Normal Distribution Z value greater than 3.49 in the z-table

Our class was given a problem where Z is a standard normal random variable and we have to look for: P(Z<6.0) and P(Z>6.0) I don't know what to do since the value is over the max in the z-table ...
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0answers
34 views

MLR $H_0 : \beta_0 \geq 1$ versus $H_a : \beta_0 < 1$

Compute the Pvalue of the test $H_0: \beta_0 \geq 1$ versus $H_a: \beta_0 < 1$. $$Y_i = \beta_0 + \beta_1 X_{i1} + \beta_2 X_{i2} + \beta_3 X_{i1}^2 + \beta_4 X_{i1} X_{i2}+\sigma Z_i, Z_i \sim N(...
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0answers
39 views

UMVUE for Exponential probabilties

Let $X_i$ be i.i.d $exp(\lambda)$ and take any $a > 0$. I want to find the UMVUE of $P(X_i < a ) = 1-\exp(-\lambda a)$. My attempt By properties of the exponential family we know $\sum_i X_i$ ...
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1answer
34 views

Construct a $95\%$ confidence interval for $5\beta_4$

Construct a $95\%$ confidence interval for $5\beta_4$. If this question were about $\beta_4$ without the $5$, I would absolutely know what to do. But I have to idea how the $5$ comes into play. I can'...
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0answers
46 views

Using properties of score function to show unbias in function and compute variance

I would like to compute the fisher indicator for this function: $$f(y;u)=\frac{1}{\sqrt{2\pi y^3}} e^{\frac{-(y-u)^2}{2u^2y}}$$ With $y>0$. I have computed the log likelihood function and the score ...
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2answers
117 views

Find the UMVUE of $\frac{\mu^2}{\sigma}$ where $X_i\sim\mathsf N(\mu,\sigma^2)$

Suppose $X_1, ..., X_4$ are i.i.d $\mathsf N(\mu, \sigma^2)$ random variables. Give the UMVUE of $\frac{\mu^2}{\sigma}$ expressed in terms of $\bar{X}$, $S$, integers, and $\pi$. Here is a ...
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1answer
38 views

Showing indepedence of two random variables when $p(x,y) = p(x) \cdot p(y)$ except a constant factor?

During a course I attend at university, I encountered the following question: Given is a probability distribution: $$p(x,y) = \lambda \eta \cdot \exp(-\lambda x - \eta y) $$ supported on $\mathbb{R}...
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1answer
33 views

Density chart of S&P500

I am examining the daily log returns of the S&P500 Index and I have negative skewness and excess kurtosis. However when I chart the density plot I am seeing positive skewness - does this seem ...
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1answer
23 views

Help understanding a probability inequality

I'm working throught Wasserman's "All of Statistics" book. When proving convergence of random variables/distributions in chapter 5, he lists the following inequality: $$F_n(x) = \mathbb{P}(X_n\le x)=\...
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1answer
138 views

Find UMVUE of $\theta$ where $f_X(x\mid\theta) =\theta(1 +x)^{−(1+\theta)}I_{(0,\infty)}(x)$

As a slight modification of my previous problem: Let $X_1, X_2, . . . , X_n$ be iid random variables having pdf $$f_X(x\mid\theta) =\theta(1 +x)^{−(1+\theta)}I_{(0,\infty)}(x)$$ where $\...
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0answers
19 views

Presenting simulation results on a research paper

I recently wrote a research paper on time series forecasting, "weights and biases initialization in ANN using multi-objective Cuckoo Search algorithm". The paper was rejected but they gave me comments ...
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2answers
51 views

How to sample for conditional probability from unknown populations

I am providing the full question as well my solution below. I'm looking for help with part (d), a simulation question. Q - Suppose there are two species of Pandas, $T_1$ and $T_2$ which are ...
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2answers
124 views

Finding UMVUE of $\theta e^{-\theta}$ where $X_i\sim\text{Pois}(\theta)$

Suppose $X_1, X_2, . . . , X_n$ are i.i.d Poisson ($\theta$) random variables, where $\theta\in(0,\infty)$. Give the UMVUE of $\theta e^{-\theta}$ I found a similar problem here. I have that the ...
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3answers
98 views

Likelihood function when $X\sim U(0,\theta)$

Let $X_1, ..., X_n$ be $i.i.d$ random variables, uniformly distributed over $(0,\theta)$. Derive the likelihood function given the sample $x_1, ..., x_n$. Answer The likelihood function is: \begin{...
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1answer
134 views

Find UMVUE of $\frac{1}{\theta}$ where $f_X(x\mid\theta) =\theta(1 +x)^{−(1+\theta)}I_{(0,\infty)}(x)$

Let $X_1, X_2, . . . , X_n$ be iid random variables having pdf $$f_X(x\mid\theta) =\theta(1 +x)^{−(1+\theta)}I_{(0,\infty)}(x)$$ where $\theta >0$. Give the UMVUE of $\frac{1}{\theta}$ ...
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1answer
73 views

Need advice concering forecasting next year based on irregular time-series - UPDATED

I need your advice with regards to the following inquiry: "Based on your observations, what could you say about the load for the same months in year 2019?" ...
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0answers
63 views

Independent and Dependent Random Variables

Please give an example of 2 dependent random variables, X and Y such that P(X < Y)=1. Again, provide an example of 2 independent random variables, X and Y such that P(X < Y)=1 For the first ...
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1answer
58 views

Sufficient statistic when $X\sim U(\theta,2 \theta)$

Let $X_1, ..., X_n$ be $i.i.d$ random variables, uniformly distributed over $(\theta,2 \theta)$. Find a sufficient statistic for $\theta$, and compute $\widehat{\theta}_{MLE}$. Answer The joint ...
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1answer
39 views

What is the Bayesian Prior Predictive distribution from two normal populations?

The question goes as follows: A shoe factory produces brown shoes and black shoes. They look the same but differ only in their weight characteristics. Brown shoes have their weight distributed as ...
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27 views

Calculating minimum of a function using Gradient Descent

I need to calculate the minimum of function : f(x) = (x − 3)2 , starting at x = 0 and with α = 1/3, by applying gradient descent. Could someone please help me here how to go about it? Found no clear ...
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1answer
79 views

CDF of Piecewise Folded Normal

I came across a problem in a Carmona's Statistical Analysis of Financial Data in R (pg. 189, Problem 3.13). The due date has passed, so now it is considered a self-study question. I am seeking a ...
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0answers
39 views

What is the probability density function of N(x;…) * N(x;…)?

Task: What is the PDF of $$ p(x) = \mathcal{N}(x;\mu_1, \sigma_1)\mathcal{N}(x;\mu_2, \sigma_2) $$ Hint: what distribution will the PDF belong to? Maybe you can simply compute the new mean and ...
3
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1answer
88 views

Linear Regression of Indicator Matrix: sum of predictions is 1

In Element of Statistical Learning, chapter 4-2 about linear regression of an indicator matrix, it is stated that the sum of predictor is equal to 1. To bring a bit of context: We have $N$ training ...
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1answer
25 views

Showing these statistics are ancillary

Let $Z_i = X_{(n)} - X_{(i)}$ for $i=1,2,\dots,n$ where $X \sim N(\mu, 1)$, and $X_{(i)}$ is the ith order statistic of the sample. I want to show $Z=(Z_1,\dots,Z_{n-1})$ are ancillary for $\mu$. My ...
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0answers
25 views

Stationarity restriction of a TGARCH process?

What is the stationarity/convergence restriction for a threshold GARCH model, TGARCH? I know that for a GARCH model: $\alpha+\beta<1$, but I'm guessing it's not that simple for a TGARCH model. ...
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0answers
13 views

A function that changes the dimension of a random vector make the random vector discrete?

One of my problems ask the question Let $\vec{x} \in \mathbb{R}^n$ be a random vector, and $g: \mathbb{R}^n \to \mathbb{R}^k$ be measurable. Then show that $g(\vec{x})$ is a discrete random vector. ...
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41 views

Conditioning to derive the distribution of function of uniform random variables

After seeing this question here, I was genuinely curious if there was a way to derive this distribution. I've attempted it below using the CDF for $Z$ and conditioning on the value of $Y$. It is ...
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0answers
14 views

Estimating a GARCH-M model with a mean-equation dummy

I'm trying to estimate a GARCH-M model with a dummy volatility variable on the mean equation, so the mean equation looks something like this: $$r_t = \mu + \lambda_1{\mkern 1mu} \sigma_t + \lambda_2 ...
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1answer
81 views

principle component analysis: help with interpretation

I'm teaching PCA to myself for some environmental data analysis. I understand the intuitive and geometric definition, but I'm not quite sure what exactly it's telling me. What exactly do the ...
0
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1answer
40 views

Probability of 3 dice with rethrown [closed]

In a game, three ordinary dice are thrown. If the first throw results in just two dice showing the same number, the die with different number if thrown again. If all numbers are different in the first ...
0
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0answers
54 views

Maximum likelihood estimator for a function containing an indicator variable

This problem is taken from the following MIT OCW course (Problem 1 question 3): https://ocw.mit.edu/courses/mathematics/18-650-statistics-for-applications-fall-2016/assignments/MIT18_650F16_PSet3.pdf ...
4
votes
3answers
88 views

Finding the distribution of a function of a normal random variable

Question: A particle's velocity $V$ is normally distributed with mean 0 and variance $\sigma^2$. The particle's energy is given by $W=m\frac{V^2}{2}$, where $m>0$ is a constant. (a) What is $\...
0
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1answer
41 views

Normal Distribution using Z - Score Rules

The distribution of heights of adult men in the U.S. is approximately normal with mean 69 inches and standard deviation 2.5 inches. Use what you know about a normal distribution and the 68-95-99.7 ...