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.

Filter by
Sorted by
Tagged with
2
votes
0answers
18 views

Small sample-size, two sample test, power calculation

We have two normally distributed random variables with means $\mu_X$ and $\mu_Y$. We test the null hypothesis $H_0:\mu_X-\mu_Y=\delta=0$, at significance level $\alpha$ in a two-tailed test. We sample ...
1
vote
1answer
7 views

Invertibility of covariance matrix when number of training examples are lesser than number of features

I was trying to study an outlier detection algorithm and realized that in case we use a multinomial Gaussian distribution to model data then the invertibility of Covariance matrix ($\sum$) is ...
0
votes
0answers
7 views

Could someone use this concrete data set to illustrate how to compute the average Gain?

Chapter 3, Page 86 of Tom M. Mitchell. Machine Learning (free) says One practical issue that arises in using GainRatio in place of Gain to select attributes is that the denominator can be zero or ...
2
votes
1answer
26 views

How to relate the distributions of these trigonometric functions of uniform variables?

$\theta_1,\theta_2$ are two independent random variables distributed uniformly in $[0,2π)$. Let $X=\cos\theta_1,Y=\cos\theta_2.$ Prove that $\frac{X+Y}{2}$ and $XY$ are equal in distribution,$\frac{X+...
0
votes
1answer
28 views

Optimizing Alpha and Beta Values for Beta-Binomial Model

I am attempting to use a beta-binomial model to predict the number of wins for each NBA basketball team for this current season. For the prior, I have pre-season win projections for each team. For ...
0
votes
0answers
15 views

OLS of VAR(2) model [closed]

Could you help me with this problem?? I'm trying to solve that for three days, but I was not able to get some solution..
2
votes
2answers
36 views

Hazard crossing for survival curve

I am very confused because people say the following pic is showing hazard crossing, but the y_aixs is actually not h(t), it is s(x) the survival probability, then how do you conclude they are showing ...
0
votes
0answers
15 views

Exercice Help! Compute the simple regression equation of Y on x for z one standard deviation above the mean [closed]

I need help with this question. The correct answer is b, but I have no idea why. Can someone explain to me how to get to this answer please? Thanks!
0
votes
1answer
13 views

True or False: The pattern of the residuals vs predicted values and the residuals vs a predictor based on a 2nd order model are exactly the same

I'm currently working on an assignment for a course in Statistics and was wondering anyones thoughts on this. My friends and I keep going back and forth on which one it is. I think it's true for the ...
0
votes
2answers
28 views

How to find expected cost when a random variable associated with that cost is normally distributed?

The question is as follows; Grear Tires has produced a new tire with a mean mileage of 36500 miles and std dev of 5000 (assume normal). Grear has offered to refund a $1 per 100 miles short of 30000. ...
2
votes
1answer
24 views

Determine if the following Markov chain is positive recurrent, null recurrent or transcient

We consider the Markov Chain with transition probabilities $$ p(i,0)=\frac{1}{i^2 +2},\qquad p(i,i+1)= \frac{i^2 +1}{i^2 +2}. $$ Determine if this Markov chain is positive recurrent, null recurrent ...
-1
votes
0answers
11 views

How to find a percentage change with multiple percentage rates? [closed]

If the height of a triangle is decreased by 40% and its base is increased by 40%, what will be effect on its area? A. 16% decrease B. 8% decrease C. No change D. 16% increase E. 8% increase
0
votes
1answer
26 views

What is the rejection region for this test?

A sample of size 16 was randomly drawn from a normal population with known standard deviation σ = 5. The sample mean is equal to 20.5. The researcher wants to run the followingz-test(α=5%): H0 :μ=...
0
votes
0answers
19 views

Show regression line passes through points $(X_1,\bar{Y_2}),(X_2,\bar{Y_2})$

Assuming a simple linear regression model , $n_1$ points are sampled at $X_1$ and $n_2$ at $X_2$ and let $\bar{Y_1} , \bar{Y_2}$ be the averages at $ X_1 , X_2$ respectively. The problem that I am ...
0
votes
0answers
21 views

Two ways to compute variance of an OLS estimator

I'm given a following problem: Considering the model $y=\beta_0+\beta_1x_1+\beta_2x_2+u$, where $E(u|x_1,x_2)=0$ and $Var(u|x_1,x_2)=\sigma^2$. We consider two estimators of $\beta_1$: $\tilde{\...
1
vote
1answer
22 views

Generalized likelihood ratio in uniform distribution

My question is to come up with the form of the GLR when testing the following: Let $X_1,\ldots, X_n$ be a random sample from a distribution with pdf $f(x;\theta,\mu) = \frac{1}{2\theta}$ if $|x-\mu|...
0
votes
0answers
14 views

Marginal distributions of a mixture of two Gaussian distributions

In the Mathematics for Machine Learning book (Section 6.4), the author introduces a mixture of two Gaussian distributions as $$0.4 \mathcal{N}\Bigg(\begin{bmatrix}10 \\ 2\end{bmatrix}, \begin{...
1
vote
0answers
14 views

Asymptotic normality: proof strategy

Given a estimator $\hat \theta$ of $\theta$, I want to show that $\sqrt{n}(\hat\theta -\theta-B)\to N(0,V_\theta)$ as $n\to\infty$, given that the limit $V_\theta$ exists and $B>0$ possibly ...
3
votes
1answer
70 views

Linearisation of Kalman filter

Assume we have the following state-space model: $$ z_{k} = \theta_{k} z_{k-1} + v_{k}\\ \theta_{k} = \phi \theta_{k-1} + w_{k}, $$ where $v_{k}$ and $w_{k}$ are independent and normal for all $k$. The ...
0
votes
1answer
34 views

Multiple linear regression, partial F-Test

I had seen many formulas out there and I hardly knew how to solve this exercise. Where it consists of: Enter link data set for the question I use $X_2,X_7\; and \;X_8$ Using the partial F test, ...
2
votes
1answer
19 views

Power of a statistical test against a compound alternative hypthosis

I was recently posed a homework question. It will be a few days before I can get in-person help from faculty, so I wanted to ask here. The question was: A medical resident doing research wants to ...
1
vote
0answers
11 views

Distribution of noncentrality parameter in noncentral t-distribution

Let $X$ be a random vector $X \sim N(0, \Sigma)$, where $\Sigma = \frac{1}{n} Id$. Find probability $$ P(|T| > a)$$ where $T$ has noncentral t-distribution with noncentrality parameter $\delta = \...
0
votes
1answer
12 views

Stochastic Models (probability)- simple symmetric random walk question

My study group and I are currently stumped on this probability question related to Stochastic models. Let {Xn} be a simple symmetric random walk (ie p = 1/2). ) Give an approximation of the ...
1
vote
0answers
30 views

Uniform distribution problems

I was confused about this problem Two emergency response units, unit A and unit B, patrol uniformly and independent a 10-km stretch of moad. An emergency incident occurs on the road and its position ...
0
votes
0answers
15 views

Use cross-validation to estimate a confidence interval for the predictions [closed]

Describe how you would use cross-validation to estimate a confidence interval for the predictions made by a machine learning algorithm.
0
votes
1answer
30 views

Simple Random Walk question

I am having trouble with these questions, I understand the rules I am meant to use, such as Markov property and independent increments yet I am having issues applying that to the question. I am also ...
2
votes
1answer
66 views

Putnam problem - guess an integer in an odd number of attempts

I have no clue how to answer this question. This is 2002 Putnam examination question B-4. B4 An integer $n$, unknown to you, has been randomly chosen in the interval $[1,2002]$ with ...
0
votes
1answer
18 views

Overfitting the validation set

I am referring to the Training / Validation / Test set for choosing a model while taking care of overfitting. Here is how the argument goes:- We train various models on the Training set. (This one ...
5
votes
1answer
64 views

Log likelihood question I need help with [closed]

Could someone explain part c) of this question to me please as it has me confused. Parts a) and b) were fine but c) is confusing me. Thanks, IS
3
votes
1answer
49 views

Sum of all probabilities is not 1 in homework

Excerpt from "A first course in Probability, 8ed" A student is taking a one-hour-time-limit makeup examination. Suppose the probability that the student will finish the exam in less than $x$ hours ...
0
votes
0answers
23 views

Distribution of sum of Laplace distributed variables [closed]

So I came across below problem: \begin{align} &Let \ x_1, \ x_2... \in L(a) \ and \ N \in Fs(p) \end{align} Hence, $N$ is First success distributed and $X$ is Laplace. Now define \begin{align} ...
1
vote
0answers
22 views

Fitting curves in Jupyter Notebook without using the fit function [closed]

For a task for school, I have to write my own fit function by using the least squares method. The problem is I don't know how to do that, specifically I don't know how to minimize my function to ...
0
votes
1answer
24 views

Conditional Poisson Process

I cannot reach a correct answer and I don't know why. I am trying to calculate this by conditioning on $N(t)=n$ and I ended up with $e^{-At(z^s)}$. However, the correct answer is $\dfrac{e^{-At(z^s)}-...
2
votes
2answers
54 views

Find the value of $\nu$ so that $n^\nu (1-X_{(n)})$ converges in distribution

Let $X_1, X_2, \cdots$ be iid. If $X_i \sim Beta(1,\beta)$, find the value of $\nu$ so that $n^\nu (1-X_{(n)})$ converges in distribution. My thoughts: Since $X_{(n)} \to 1$ in probability, I was ...
1
vote
1answer
49 views

How and when to use paired t-test?

In a study of the effectiveness of weight loss programs, 47 subjects who were overweight, at least 20% participated in a group support program for 10 weeks. The weight of each subject was taken at the ...
8
votes
2answers
335 views

How can I find the distribution for the number of years until the first year's rainfall is exceeded for the first time?

Let $X_1, X_2, \cdots$ be jointly continuous and independently distributed with marginal pdf $f(x)$, where each $X_i$ represents annual rainfall at a given location. How can I find the distribution ...
0
votes
0answers
12 views

Jointly Sufficient Statistics [closed]

Please Help me solve this homework question, I have been struggling with it for a couple of days already
0
votes
0answers
18 views

Logistic regression threshold of independent variable [closed]

I have a logistic regression model for atmosphere toxicity where the dependent variable is toxic (1) if the atmosphere of a country is toxic and non toxic (0) and the independent variable is the ...
1
vote
1answer
48 views

Prove that $E[e^{2(m−1)X^2}]\le m$

I'm reading "Understanding Machine Learning: From theory to algorithms". The problem is as follows, which is Exercise 31.1 of the book on page 416. Let $X$ be a random variable that satisfies $P[X \...
2
votes
1answer
47 views

Kalman filter: asymptotic of state estimate

Assume we have a linear state-space model: $$ z_{k} = Hx_{k} + v_{k}\\ x_{k} = F x_{k-1} + Bu + w_{k}, $$ where $u$ is some control variable (constant intercept is the simplest case). Kalman filter, ...
0
votes
0answers
30 views

Introduction to Statistical Learning

For those that have read the book Introduction to Statistical Learning, I'm having a problem understanding a certain line: I can't seem to understand the derivation from the first equality to the ...
1
vote
0answers
34 views

How can I find the marginal distributions of $\frac{X_1}{X_1+X_2}$ and $\frac{X_2}{X_1+X_2}$? [duplicate]

Let $X_1 \sim Gamma(\alpha_1,1)$ and $X_2 \sim Gamma(\alpha_2,1)$ be independent random variables. How can I find the marginal distributions of $\frac{X_1}{X_1+X_2}$ and $\frac{X_2}{X_1+X_2}$? By ...
0
votes
0answers
22 views

Expected wait time in two different types of banks?

Problem Statement There are two banks both have 100 people inside who want to cash their check and must do so by talking to the teller. In both banks, there are 10 tellers. In Bank 1 all 100 people ...
0
votes
0answers
21 views

linear dependence and 0/1 features in ML

This question is in the context of machine learning. I don't know much statistics (I'm just a software engineer who finds machine learning cool), and I was confused by something I just read in a book. ...
0
votes
0answers
17 views

Relationship between repeated measures ANOVA and random / mixed effect models

This is not concrete question but more a conceptual one, although a pretty low-level one I guess: When I was taught ANOVAs the 'classical' sums of squares way, I was told that repeated measures can ...
1
vote
0answers
14 views

Testing of Hypothesis using Anova

Let $y_1=\alpha_1+e_1,y_2=2\alpha_1-\alpha_2+e_2$ and $y_3=\alpha_1+2\alpha_2+e_3$ where $e_i \sim N(0,\sigma^2)$ iid for $i=1,2,3$. Find the least square estimate of $\alpha_1$ and $\alpha_2$. Derive ...
2
votes
2answers
252 views

Poisson Distribution vs Binomial Distribution

I have a specific problem here. It is stated as: "Assume that 30% of students in a university take public transportation daily to commute to their college. Suppose 10 of the students are randomly ...
0
votes
1answer
22 views

Why autocovariance is 0?

First of all I am a newbie in stats, trying to study statistics, and somehow I find 'time series' way more difficult than other topics. I am confused with the “autocovariance”. In the book "Time ...
-1
votes
1answer
59 views

How can I derive mathematical posterior predictive distribution calculation steps for beta prior and binomial likelihood

I would like to know the mathematical calculation step by step processes with beta prior and binomial likelihood for posterior predictive distribution.
1
vote
0answers
25 views

Asymptotic Distribution of Wald test

If $X_1 , X_2 , ... X_n$ are iid and they have the same pdf $f_θ(x)$ . Consider testing $H_0 : θ= θ_0$ vs $H_n : θ = θ_0 +Δ * n^{-1/2}$. where , $Δ = (Δ_1 , Δ_2 , .. Δ_k)^T$ We want to find the ...