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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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13 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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30 views

Show that for some $\delta$ $\in [0,1]$, $\frac{1}{n+m}\sum^{n+m}_{i=1}Var(\epsilon_i)= \delta \sigma^2_1+(1-\delta)\sigma^2_2.$ [on hold]

Consider the linear regression setting where we have $n$ sample points with $ \epsilon \sim N(0, \sigma ^2_1)$ and $m$ sample points with $\epsilon \sim N (0, \sigma^2_2)$ (a) Show that for some $\...
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1answer
39 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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11 views

Batch versus stochastic gradient descent in R [on hold]

Develop two learning algorithms: batch and stochastic gradient descent for this problem on the Auto dataset from ISLR. Try to find the two best input features for predicting the output mpg. Compare and ...
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35 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 ...
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20 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
18 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
11 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
12 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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35 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
7 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
31 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 ...
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1answer
37 views

Probability of 3 dice with rethrown [on hold]

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 ...
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0answers
10 views

Obtaining the joint distribution of state and observations for a Linear Gaussian State-Space Model

I am trying to follow the paper Smoothing Algorithms for State-Space Models. In section IV the authors summarize the Kalman Filter for a linear Gaussian model. The set up is as follows: Let $X_t$ ...
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0answers
30 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 ...
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3answers
71 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 $\...
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1answer
32 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 ...
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1answer
48 views

Probability of passengers exiting an elevator

I'm not sure if I'm doing this problem correctly. The question is to ...
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2answers
71 views

Find an estimator for $\theta$ when PDF is:

Find an estimator for $\theta$ when PDF is: $$ f(x) =(1-\theta)\mathbb{I}_{[-1/2,0)}(x)+ (1+\theta)\mathbb{I}_{(0,1/2]}(x). $$ I know that one way is to write the likelihood function then do Log-...
3
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1answer
80 views

Distribution of sum of exponentials

Let $X_1$ and $X_2$ be independent and identically distributed exponential random variables with rate $\lambda$. Let $S_2 = X_1 + X_2$. Q: Show that $S_2$ has PDF $f_{S_2}(x) = \lambda^2 x \text{e}^{-...
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34 views

Check Computation of MME and MLE

Let $X_1$, . . . , $X_n$ be i.i.d random variables having pdf $$f(x\mid\theta) = (\theta+ 1)x^{\theta}I_{(0,1)}(x)$$ where $\theta \gt−1$ (a) Give a MME of $\theta$ based on the first ...
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45 views

Baseball game duration distributions [closed]

The following represents the game time, in minutes, for the Major League Baseball games played in the United States between March 30 and April 9, 2003, sorted by league. Compare the times between the ...
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24 views

Constant Terms in Linear Projection

In my time series textbook, it says, "Let $Y_i$ and $Y_j$ be two dependent variables in a time series process, e.g. $Y_{t+1}=\phi Y_{t}+\epsilon_{t+1}$, where $\phi$ is a constant coefficient. If a ...
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1answer
38 views

Obtaining posteriors for multivariate Normal mixture models

So I want to fit a mixture model $$f(y) = \pi_1 f_1 (y) + \pi_2 f_2 (y)$$ where $\pi_k = P(S = k)$ and $S_i$ is a latent unobserved variable. I assume that, conditional on $S=k$, we have the model ...
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16 views

Likelihood Construction for Censored Data

I am trying to understand the Expectation-Maximization algorithm, and was trying to read through this paper by Park and Lee. In section 2, "Likelihood Construction for Censored Data", they mention the ...
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42 views

MLE Where Observations are Frequency Counts

Consider independent trials, such that for each trial a type 1 outcome will occur with probability $p_1= \frac{1}{2} + \frac{\theta}{4}$ a type 2 outcome will occur with probability $p_2= ...
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1answer
72 views

Beta Distribution and how it is related to this question

Let $f(x) = k(\sin x)^5(1-\sin x)^7$ if $0 \lt x \lt \pi/2$ and $0$ otherwise. Find the value of $k$ that makes $f(x)$ a density function. I'm struggling to understand how this relates to the Beta ...
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1answer
24 views

Question on complete sufficient statistic

In my textbook, I have this example which says $X\sim U(0,\theta)$ we show that the family of PDFs of X is complete. We need to show that $E(g(x))=\int_{0}^{\theta}\frac{1}{\theta}g(x)dx=0\ \ \ \ \ \...
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1answer
17 views

Test for exogenity

I have a few questions about this alternative test for exogeneity. The first question is about the auxiliary regression? I have tried to google it and still don't understand the meaning of it. What ...
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24 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 ...
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1answer
56 views

Understanding bayesian model code from chapter 4 of “Statistical rethinking”

I'm trying to learn bayesian statistics from "Statistical rethinking" by Richard McElreath. In chapter 4, a model with Gaussian distribution of heights is introduced: $h_i \sim N(\mu, \sigma)$ $\mu \...
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1answer
86 views

MLE of $f(x\mid\theta) = \theta x^{\theta−1}e^{−x^{\theta}}I_{(0,\infty)}(x)$

$X_1$, $X_2$, and $X_3$ are i.i.d random variables having pdf $$f(x\mid\theta) = \theta x^{\theta−1}e^{−x^{\theta}}I_{(0,\infty)}(x)$$ where $\theta \gt0$. Using the observed values of these ...
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2answers
25 views

Prove minimum argument of square error function is equal to expected value

Consider a probability density function $f(x)$ defined over the interval $[a,b]$ where $-\infty<a<b<\infty$. The square error function is defined as $J(y)=\int_a^b (x-y)^2f(x)dx$. The ...
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1answer
124 views

Maximum Likelihood Estimate for a likelihood defined by parts

Suppose $X_1$, . . . , $X_n$ are i.i.d random variables having pdf $$ f(x\mid\theta)= \begin{cases} \frac{4}{\theta}-\frac{4x}{\theta^2} & \frac{\theta}{2} \lt x \lt \theta \\ \frac{4x}{...
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16 views

How to get the sample size to meet a certain power criterion

How should I find n (sample size) in the following equation? $$p(Z<=-0.01\sqrt n-1.64)+p(Z>=-0.01\sqrt n+1.64)=0.95$$
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24 views

Regression equation passing through the origin

I'm going through Multiple Choice Questions of Basic Econometrics by Gujarati. There is a question which states that: It is a simple two-variable regression: Any regression equation written in its ...
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27 views

Real world stat practice

I've been taking courses in Applied Statistics and through the homework problems I get to practice what I've learned. But, I find that the practice problems are always a bit contrived to fit within ...
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0answers
22 views

Cross Validation, Variable Definition [closed]

I am working on a 10-fold cross validation problem, and am having an issue with part of my code. Specifically, I'm having a problem with my "for (1 in nfold)" argument, and with the variable length of ...
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1answer
23 views

Systematic way to determine if a model is linear or nonlinear? [duplicate]

Determine whether the following models are linear, intrinsically linear, or nonlinear (disregard the error structure): $y=\beta_0+\beta_1 x_1 +\beta_2 x_2^{\beta_3}+\epsilon$ $y=\beta_1 + \...
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15 views

Random dicrete variable in a biased coin

A coin is biased so that the probability of obtaining a tail is $0.75$.the coin is tossed four times and the random variable x is the number of tails obtained. Find i) $E[2X]$ ii) $Var(3X)$ In the ...
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29 views

Tutorial question on min number of sample size for confidence interval

I'm stuck with this question from my tutorial (and there is no worked solution), and I can't seem to get the correct answer of 411. There were 904 new Subway Restaurants franchises opened during 2002....
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1answer
20 views

Multiplying two event with probability density function, is it possible?

in my exercise, $X$ is the size of a tree trunk, and $X$ follows a normal distribution $\mathcal{N}(9,0.4)$, we want to know $P(8.8\le X\le11.2)$ So I though that I could do this: $P(8.8\le X \le11.2)...
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25 views

Probability, conditional probability question

Can someone help me with this probability question? Suppose there are two coins A and B. A shows heads with probability $p$ and B shows heads with probability $v$. First select a coin at random, with ...
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1answer
19 views

exponential distribution [closed]

The time it takes for a team to complete a certain task has an exponential distribution with mean equals to 80 hours. Given that the task was not completed in 50 hours, what is the probability that ...
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1answer
126 views

Estimating the Parameters for $y=\beta_1 e^{\beta_2 x}+\beta_3 z+\epsilon$

I have the model $$y=\beta_1 e^{\beta_2 x}+\beta_3 z+\epsilon$$ where $z$ is an indicator variable. I need to obtain estimates from linear regression to get initial values for the parameters. Then I ...
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2answers
48 views

Variance of new $y$ - $\hat{y}$

Suppose we have a linear regression $Y = X\beta + e$, and we have new data come in. $Y_{new} = X_{new}\beta + e_{new}$. And we know that $Y_{new}$ and $\hat{Y_{new}}$ are independent. So $Var(Y_{new} -...
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0answers
9 views

significance test by confidence interval in complete randomization design(CRD)

In page 91 of this book, Design and analysis of experiments by Montgomery , the author stated after Equation (3.30) that: Clearly, if the confidence interval in Equation 3.30 includes zero, ...
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1answer
19 views

Number of ways to place subset of balls between balls

If there are 5 balls : b1, b2, b3, b4, b5 that have been randomly arranged what is the probability 1 ball occurs between b1 and b2 three balls occur between b1 and b2 Here is my attempt : ...
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1answer
33 views

How can I test whether an individual treatment mean significant or not?

Suppose by complete randomization design, I reach into a decision that there is at least difference between two treatment means. That is, my hypothesis is $$H_o:\mu_1=\mu_2=\mu_3=\mu_4=\mu_5$$ $$...
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2answers
43 views

How to remove the effect of one variable by using linear model residuals

My data set has species with a number of morphological variables, including body mass: ...