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Questions tagged [self-study]

A routine exercise designed to test one's knowledge; often 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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146 votes
19 answers
125k views

Books for self-studying time series analysis?

I started by Time Series Analysis by Hamilton, but I am lost hopelessly. This book is really too theoretical for me to learn by myself. Does anybody have a recommendation for a textbook on time ...
90 votes
3 answers
193k views

An example: LASSO regression using glmnet for binary outcome

I am starting to dabble with the use of glmnet with LASSO Regression where my outcome of interest is dichotomous. I have created a small mock data frame below: <...
Matt Reichenbach's user avatar
71 votes
6 answers
232k views

Find expected value using CDF

I'm going to start out by saying this is a homework problem straight out of the book. I have spent a couple hours looking up how to find expected values, and have determined I understand nothing. ...
styfle's user avatar
  • 937
64 votes
4 answers
42k views

How are regression, the t-test, and the ANOVA all versions of the general linear model?

How are they all versions of the same basic statistical method?
Amahabirsingh's user avatar
60 votes
7 answers
8k views

Interview question: If correlation doesn't imply causation, how do you detect causation?

I got this question: If correlation doesn't imply causation, how do you detect causation? in an interview. My answer was: You do some form of A/B testing. The interviewer kept prodding me for ...
Akaike's Children's user avatar
57 votes
13 answers
8k views

Visually interesting statistics concepts that are easy to explain

I noticed on Math Stack Exchange a terrific thread which highlighted a number of very visually interesting math concepts. I would be curious to see graphics/gifs which anyone has that very clearly ...
54 votes
1 answer
118k views

What is the derivative of the ReLU activation function?

What is the derivative of the ReLU activation function defined as: $$ \mathrm{ReLU}(x) = \mathrm{max}(0, x)$$ What about the special case where there is a discontinuity in the function at $x=0$?
Tom Hale's user avatar
  • 2,601
51 votes
4 answers
12k views

Taking the expectation of Taylor series (especially the remainder)

My question concerns trying to justify a widely-used method, namely taking the expected value of Taylor Series. Assume we have a random variable $X$ with positive mean $\mu$ and variance $\sigma^2$. ...
agronskiy's user avatar
  • 695
48 votes
4 answers
180k views

McFadden's Pseudo-$R^2$ Interpretation

I have a binary logistic regression model with a McFadden's pseudo R-squared of 0.192 with a dependent variable called payment (1 = payment and 0 = no payment). What is the interpretation of this ...
Matt Reichenbach's user avatar
48 votes
7 answers
9k views

Why shouldn't the denominator of the covariance estimator be n-2 rather than n-1?

The denominator of the (unbiased) variance estimator is $n-1$ as there are $n$ observations and only one parameter is being estimated. $$ \mathbb{V}\left(X\right)=\frac{\sum_{i=1}^{n}\left(X_{i}-\...
MYaseen208's user avatar
  • 2,729
48 votes
8 answers
7k views

Pitfalls in time series analysis

I am just starting out self-learning in time series analysis. I have noticed that there are a number of potential pitfalls that are not applicable to general statistics. So, building on What are ...
45 votes
6 answers
45k views

Why does the variance of the Random walk increase?

The random walk that is defined as $Y_{t} = Y_{t-1} + e_t$, where $e_t$ is white noise. Denotes that the current position is the sum of the previous position + an unpredicted term. You can prove that ...
Isbister's user avatar
  • 779
45 votes
1 answer
39k views

Derivation of change of variables of a probability density function?

In the book pattern recognition and machine learning (formula 1.27), it gives $$p_y(y)=p_x(x) \left | \frac{d x}{d y} \right |=p_x(g(y)) | g'(y) |$$ where $x=g(y)$, $p_x(x)$ is the pdf that ...
dontloo's user avatar
  • 16.5k
45 votes
6 answers
56k views

What's the difference between logistic regression and perceptron?

I'm going through Andrew Ng's lecture notes on Machine Learning. The notes introduce us to logistic regression and then to perceptron. While describing Perceptron, the notes say that we just change ...
GrowinMan's user avatar
  • 961
43 votes
5 answers
24k views

LDA vs word2vec

I am trying to understand what is similarity between Latent Dirichlet Allocation and word2vec for calculating word similarity. As I understand, LDA maps words to a vector of probabilities of latent ...
Piotr Migdal's user avatar
  • 5,786
42 votes
3 answers
60k views

How to take derivative of multivariate normal density?

Say I have multivariate normal $N(\mu, \Sigma)$ density. I want to get the second (partial) derivative w.r.t. $\mu$. Not sure how to take derivative of a matrix. Wiki says take the derivative ...
user1061210's user avatar
  • 1,085
42 votes
8 answers
115k views

Looking for a good and complete probability and statistics book

I never had the opportunity to visit a stats course from a math faculty. I am looking for a probability theory and statistics book that is complete and self-sufficient. By complete I mean that it ...
41 votes
2 answers
33k views

What is the difference between censoring and truncation?

In the book Statistical Models and Methods for Lifetime Data , it is written : Censoring: When an observation is incomplete due to some random cause. Truncation: When the incomplete nature of ...
ABC's user avatar
  • 1,705
39 votes
9 answers
9k views

What is the relationship between $Y$ and $X$ in this plot?

What is the relationship between $Y$ and $X$ in the following plot? In my view there is negative linear relationship, But because we have a lot of outliers, the relationship is very weak. Am I right? ...
PSS's user avatar
  • 843
39 votes
4 answers
12k views

Can anyone clarify the concept of a "sum of random variables"

In my probability class the terms "sums of random variables" is constantly used. However, I'm stuck on what exactly that means? Are we talking about the sum of a bunch of realizations from a random ...
Gosset's user avatar
  • 555
39 votes
3 answers
9k views

What does the Akaike Information Criterion (AIC) score of a model mean?

I have seen some questions here about what it means in layman terms, but these are too layman for for my purpose here. I am trying to mathematically understand what does the AIC score mean. But at ...
caveman's user avatar
  • 2,711
37 votes
5 answers
5k views

Will the fact that my Italian son is going to attend a primary school change the expected number of Italian children to be present in his class?

This is a question stemming from a real-life situation, for which I have been genuinely puzzled about its answer. My son is due to start primary school in London. As we are Italian, I was curious to ...
jj90213's user avatar
  • 445
35 votes
1 answer
22k views

Difference between Hidden Markov models and Particle Filter (and Kalman Filter)

Here is my old question I would like to ask if someone knows the difference (if there is any difference) between Hidden Markov models (HMM) and Particle Filter (PF), and as a consequence Kalman ...
user5584748's user avatar
33 votes
2 answers
86k views

How to know if a data follows a Poisson Distribution in R?

I am an undergrad student and I have a project for my probability class. Basically, I have a dataset about the hurricanes that impacted my country for a series of years. In my probability Book, (...
Shariff's user avatar
  • 431
33 votes
4 answers
124k views

probability of one random variable being greater than another

Using the normal distribution. Let $X \sim N(1, 2)$ and $Y \sim N(2, 3)$ where $N(\mu, \sigma^2)$ denotes the normal distribution with mean $\mu$ and variance $\sigma^2$. $X$ and $Y$ are independent. ...
dataznkid1's user avatar
32 votes
7 answers
6k views

How can I prove mathematically that the mean of a distribution is the measure that minimizes the variance?

According to the Page 87 in Kruschke: Doing Bayesian Data Analysis, the author says that the mean of a distribution is a value that minimizes the variance of a probability distribution, for example, a ...
xabzakabecd's user avatar
  • 3,495
32 votes
1 answer
21k views

Proof of LOOCV formula

From An Introduction to Statistical Learning by James et al., the leave-one-out cross-validation (LOOCV) estimate is defined by $$\text{CV}_{(n)} = \dfrac{1}{n}\sum\limits_{i=1}^{n}\text{MSE}_i$$ ...
Clarinetist's user avatar
  • 5,027
31 votes
8 answers
121k views

Two dice rolls - same number in sequence

I am currently studying Statistical Inference class on Coursera. In one of the assignments, the following question comes up. ...
Rishabh Sagar's user avatar
31 votes
7 answers
7k views

What are the branches of statistics?

In mathematics, there are branches such as algebra, analysis, topology, etc. In machine learning there is supervised, unsupervised, and reinforcement learning. Within each of these branches, there are ...
31 votes
1 answer
73k views

Pdf of the square of a general normal random variable

Given a known Gaussian distribution, $X \sim \mathcal N(\mu_x, \sigma_x^2)$, how does one determine the the distribution of $Y$ if $Y = X^2$?
Melye77's user avatar
  • 421
31 votes
3 answers
66k views

Why is nls() giving me "singular gradient matrix at initial parameter estimates" errors?

I have some basic data on emission reductions and cost per car: ...
Amanda's user avatar
  • 441
31 votes
4 answers
2k views

Is the result of an exam a binomial?

Here is a simple statistics question I was given. I'm not really sure I understand it. X = the number of aquired points in an exam (multiple choice and a right answer is one point). Is X binomial ...
Paul's user avatar
  • 445
30 votes
1 answer
2k views

What are the classical notations in statistics, linear algebra and machine learning? And what are the connections between these notations?

When we read a book, understanding the notations plays a very important role of understanding the contents. Unfortunately, different communities have different notation conventions for the formulation ...
29 votes
4 answers
5k views

Self study vs a taught education?

There is a question with similar intent on programmers.SE. That question has some quite good answers, but the general theme seems to be that without self study, you get no-where. Obviously there are ...
29 votes
2 answers
10k views

Hidden Markov Model vs Markov Transition Model vs State-Space Model...?

For my master's thesis, I am working on developing a statistical model for the transitions between different states, defined by serological status. For now, I won't give too many details into this ...
Ryan Simmons's user avatar
  • 1,883
28 votes
6 answers
6k views

Is a neural network essential for deep learning?

I received preliminary materials on deep learning in my class. It was written as follows. This raised me the question of the basic meaning of the word deep learning. Deep learning is a machine ...
desertpureolive's user avatar
28 votes
1 answer
11k views

Construction of Dirichlet distribution with Gamma distribution

Let $X_1,\dots,X_{k+1}$ be mutually independent random variables, each having a gamma distribution with parameters $\alpha_i,i=1,2,\dots,k+1$ show that $Y_i=\frac{X_i}{X_1+\cdots+X_{k+1}},i=1,\dots,k$,...
Argha's user avatar
  • 2,110
28 votes
5 answers
45k views

How to test if my distribution is multimodal?

When I plot a histogram of my data, it has two peaks: Does that mean a potential multi-modal distribution? I ran the dip.test in R (...
user1260391's user avatar
28 votes
3 answers
31k views

Interpreting plot of residuals vs. fitted values from Poisson regression

I am trying to fit data with a GLM (poisson regression) in R. When I plotted the residuals vs the fitted values, the plot created multiple (almost linear with a slight concave curve) "lines". What ...
jocelyn's user avatar
  • 281
28 votes
3 answers
6k views

Basic Simpson's paradox

I have a question about something that my statistics teacher said about the following problem: There are two hospitals named Mercy and Hope in your town. You must choose one of these in which to ...
swiecki's user avatar
  • 383
28 votes
1 answer
8k views

Why LKJcorr is a good prior for correlation matrix?

I´m reading the chapter 13 "Adventures in Covariance" in the (superb) book Statistical Rethinking by Richard McElreath where he presents the following hierarchical model: (...
xboard's user avatar
  • 1,204
26 votes
1 answer
47k views

Hat matrix and leverages in classical multiple regression

What is Hat matrix and leverages in classical multiple regression? What are their roles? And Why do use them? Please explain them or give satisfactory book/ article references to understand them.
1190's user avatar
  • 1,070
26 votes
9 answers
55k views

How is interpolation related to the concept of regression?

Explain briefly What is meant by interpolation.How is it related to the concept of regression? interpolation is art of reading between the lines of a table and in elementary mathematics the term ...
Argha's user avatar
  • 2,110
26 votes
2 answers
28k views

How to understand that MLE of Variance is biased in a Gaussian distribution?

I'm reading PRML and I don't understand the picture. Could you please give some hints to understand the picture and why the MLE of variance in a Gaussian distribution is biased? formula 1.55: $$ \mu_{...
ningyuwhut's user avatar
25 votes
1 answer
34k views

A proof for the stationarity of an AR(2)

Consider a mean-centred AR(2) process $$X_t=\phi_1X_{t-1}+\phi_2X_{t-2}+\epsilon_t$$ where $\epsilon_t$ is the standard white noise process. Just for sake of simplicity let me call $\phi_1=b$ and $\...
Marco's user avatar
  • 453
25 votes
2 answers
220k views

How do I calculate the variance of the OLS estimator $\beta_0$, conditional on $x_1, \ldots , x_n$?

I know that $$\hat{\beta_0}=\bar{y}-\hat{\beta_1}\bar{x}$$ and this is how far I got when I calculated the variance: \begin{align*} Var(\hat{\beta_0}) &= Var(\bar{y} - \hat{\beta_1}\bar{x}) \\ &...
M T's user avatar
  • 621
25 votes
2 answers
45k views

How do you "control" for a factor/variable?

To my understanding, "Control" can have two meanings in statistics. Control group: In an experiment, no treatment is given to the member of the control group. Ex: Placebo vs Drug: You give drugs to ...
leviathan's user avatar
  • 915
24 votes
2 answers
7k views

What is a distribution over functions?

I am reading a textbook Gaussian Process for Machine Learning by C.E. Rasmussen and C.K.I. Williams and I am having some trouble understanding what does distribution over functions mean. In the ...
kamilazdybal's user avatar
23 votes
3 answers
15k views

Simulating draws from a Uniform Distribution using draws from a Normal Distribution

I recently purchased a data science interview resource in which one of the probability questions was as follows: Given draws from a normal distribution with known parameters, how can you simulate ...
wellington's user avatar
23 votes
3 answers
143k views

Expected number of tosses till first head comes up

Suppose that a fair coin is tossed repeatedly until a head is obtained for the first time. What is the expected number of tosses that will be required? What is the expected number of tails that will ...
nicole900's user avatar
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