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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.

16
votes
1answer
7k views

Question on how to normalize regression coefficient

Not sure if normalize is the correct word to use here, but I will try my best to illustrate what I am trying to ask. The estimator used here is least squares. Suppose you have $y=\beta_0+\beta_1x_1$, ...
23
votes
3answers
6k views

Functions of Independent Random Variables

Is the claim that functions of independent random variables are themselves independent, true? I have seen that result often used implicitly in some proofs, for example in the proof of independence ...
45
votes
4answers
27k 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?
10
votes
3answers
3k views

$P[X=x]=0$ when $X$ is continuous variable

I know that for continuous variable $P[X=x]=0$. But i can't visualize that if $P[X=x]=0$, there is infinite number of possible $x$'s. And also why do their probabilities get infinitely small ?
34
votes
6answers
4k 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}-\...
17
votes
4answers
3k 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 ...
14
votes
1answer
4k 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 $Y_i=\frac{X_i}{X_1+\cdots+X_{k+1}},i=1,\dots,k$,...
24
votes
3answers
20k 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 ...
6
votes
1answer
5k views

Interpretation of interaction term

I have a model: $$ \ln({\rm earnings}) = a+b_1{\rm female}+b_2{\rm white}+b_3{\rm female}\times{\rm white} $$ ${\rm female}$ and ${\rm white}$ are dummy variables. I have interpreted $b_1$ and $b_2$: ...
27
votes
3answers
4k views

Which hospital should be chosen? One has a higher success rate, but the other has a higher overall success rate

I have a question about something that my statistics teacher said about the following problem. My question isn't even about the occurrence of Simpson's paradox in this situation. My question is simply ...
6
votes
2answers
2k views

Risk of extinction of Schrödinger's cats

I am interested in how uncertainty can be accounted for when considering the risk of extinction of a species. Forgive me for extending a rather tired thought experiment, but at least it's familiar ...
17
votes
3answers
27k 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: ...
12
votes
2answers
37k views

In left skewed data, what is the relationship between mean and median?

I think the median $\leq$ mean. Is this the case?
32
votes
3answers
32k 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 ...
38
votes
9answers
4k 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? ...
5
votes
1answer
234 views

Probability associated with experiencing all outcomes

Assume last year I commuted to work in a taxi, and suppose if there are $n$ taxis in the fleet I used. If I took one of these taxis every trip at random and with replacement, then what is the number ...
39
votes
3answers
19k views

A generalization of the Law of Iterated Expectations

I recently came across this identity: $$E \left[ E \left(Y|X,Z \right) |X \right] =E \left[Y | X \right]$$ I am of course familiar with the simpler version of that rule, namely that $E \left[ E \...
39
votes
4answers
6k 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 $\sigma^2$. ...
13
votes
1answer
21k 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 ...
17
votes
4answers
23k 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 (...
6
votes
3answers
27k views

Prove F test is equal to T test squared

I need to show that F test is equal to T test squared, when the T test is for 2 independent groups and assuming variances are equal. I know that $F=\frac{MSB}{MSW}=\frac{SSB/k-1}{SSW/N-K}$ and I know ...
15
votes
2answers
7k 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 ...
5
votes
1answer
7k views

How To Write Seasonal ARIMA model mathematically

I'm trying to write a seasonal ARIMA model ARIMA(1,0,3)(1,2,0) period 5 mathematically but I don't seem to be able to follow what this resource is saying otexts arima The example they use is ARIMA(1,...
13
votes
1answer
6k views

MLE of the location parameter in a Cauchy distribution

After centering, the two measurements x and −x can be assumed to be independent observations from a Cauchy distribution with probability density function: $f(x :\theta) = $ $1\over\pi (1+(x-\...
11
votes
1answer
3k views

Proof that if higher moment exists then lower moment also exists

The $r$-th moment of a random variable $X$ is finite if $$ \mathbb E(|X^r|)< \infty $$ I am trying to show that for any positive integer $s<r$, then the $s$-th moment $\mathbb E[|X^s|]$ is ...
5
votes
3answers
4k views

what is the meaning of 'tail' of kurtosis?

There are two kurtosis types : positive(leptokurtic) and negative(platykurtic). leptokurtic is heavy tailed, and platykurtic is thin tailed. But leptokurtic is more thinner and pointy than platykurtic ...
6
votes
2answers
4k views

Proof/Derivation of Residual Sum of Squares (Based on Introduction to Statistical Learning)

On page 19 of the textbook Introduction to Statistical Learning (by James, Witten, Hastie and Tibshirani--it is freely downloadable on the web, and very good), the following is stated: Consider a ...
4
votes
1answer
285 views

Sufficient statistics for Uniform (-θ,θ)

So, I know that $\max(-X_{(1)},X_{(n)})$ is a sufficient statistic for the parameter θ. But can I also say that $(X_{(1)},X_{(n)})$ are jointly sufficient for the paramether θ? In other words, can a ...
1
vote
1answer
4k views

Decision boundaries and Gaussian density functions

This is for my hw, and if anyone can solve the first part of the question it will be great. Here is the question: Assume a two-class problem with equal a priori class probabilities and Gaussian class-...
5
votes
2answers
2k views

Example where a simple correlation coefficient has a sign opposite to that of the corresponding partial correlation coefficient

Give some examples where a simple correlation coefficient has a sign opposite to that of the corresponding partial correlation coefficient and comment on it. It is a question from an examination ...
73
votes
3answers
138k 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: <...
31
votes
4answers
77k 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. ...
24
votes
4answers
79k views

McFadden's Pseudo-R2 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 ...
35
votes
3answers
4k 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 ...
24
votes
5answers
18k 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 ...
17
votes
9answers
2k views

Reference Request: Generalized Linear Models

I am looking for an introductory to intermediate level book on Generalized Linear Models. Ideally, in addition to the theory behind the models, I would want it to include applications and examples in ...
8
votes
2answers
27k views

Anova from R output interpretation

I have a question on how a statistician would normally interpret an anova output. Say I have anova output from R. ...
4
votes
1answer
1k views

Partitioning sum of squares

I am working through the proof for partitioning sum of squares. Could someone please explicitly explain how one goes from line 14 to 15 in the following PDF: http://capone.mtsu.edu/dwalsh/4380/...
3
votes
2answers
813 views

Identity in Simple Linear Model

I'm working on this identity $$\sum_{i-1}^n (y_i - \hat {\beta_0} - \hat {\beta_1}x_i)^2 = \sum_{i=1}^n y_i^2 - \hat {\beta_0}\sum_{i=1}^n y_i - \hat {\beta_1} \sum_{i=1}^n x_iy_i$$ I have these ...
4
votes
3answers
2k views

Normal distribution probability

The annual returns on stocks and treasury bonds over the next 12 months are uncertain. Suppose that these returns can be described by normal distributions with stocks having a mean of 15% and a ...
5
votes
2answers
505 views

Is the normal distribution a better approximation to the binomial distribution with proportions near or far from 0.5?

From the Online Stat Book: I don't understand this: The accuracy of the approximation depends on the values of N and π. A rule of thumb is that the approximation is good if both Nπ and N(1-π) ...
5
votes
1answer
2k views

Inverse function for a non-decreasing CDF

For a CDF that is not strictly increasing, i.e. its inverse is not defined, define the quantile function $$F^{-1} (u) =\inf \{x: F(x) \geq u \},\quad 0<u<1. $$ Where U has a uniform $(0,1)$ ...
14
votes
8answers
25k 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 ...
15
votes
2answers
101k 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}) \\ &...
9
votes
1answer
1k views

Least angle regression keeps the correlations monotonically decreasing and tied?

I'm trying to solve a problem for least angle regression (LAR). This is a problem 3.23 on page 97 of Hastie et al., Elements of Statistical Learning, 2nd. ed. (5th printing). Consider a regression ...
15
votes
4answers
2k views

Classic linear model - model selection

I have a classic linear model, with 5 possible regressors. They are uncorrelated with one another, and have quite low correlation with the response. I have arrived at a model where 3 of the regressors ...
10
votes
2answers
5k views

What are complete sufficient statistics?

I have some trouble understanding complete sufficient statistics? Let $T=\Sigma x_i$ be a sufficient statistic. If $E[g(T)]=0$ with probability 1, for some function $g$, then it is a complete ...
11
votes
4answers
776 views

What causes lasso to be unstable for feature selection?

In compressed sensing, there is a theorem guarantee that $$\text{argmin} \Vert c \Vert_1\\ \text{subject to } y = Xc $$ has a unique sparse solution $c$ (See appendix for more details). Is there a ...
8
votes
1answer
28k views

Expected Value and Variance of Estimation of Slope Parameter $\beta_1$ in Simple Linear Regression

I am reading a text, "Probability and Statistics" by Devore. I am looking at 2 items on page 740: the expected value and variance of the estimation of $\beta_1$, which is the slope parameter in the ...
7
votes
3answers
2k views

Conditional Expected Value of Product of Normal and Log-Normal Distribution

Could someone please provide the answer and steps to solve this expression? \begin{eqnarray*} E\left[\left.\left(e^{X}Y+k\right)\right|\left.\left(e^{X}Y+k\right)>0\right]\right. \end{eqnarray*} $...