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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E(SN) for aggregate claim amount S, S=X1+...+Xn, Xi are iid

Consider the following model for aggregate claim amounts $S$: $S=X_{1}+X_{2}+...+X_{N}$ where the $X_{i}$ are independent, identically distributed random variables representing individual claim ...
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Probability of a full-time draw in a basketball match

I got the following question : How to estimate roughly the probability of a full-time draw in an even basketball game with 200 in expected points? In a very simple model we can assume that there are ...
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When does a difference in means not capture the true treatment effects vs a regression with pre-treatment controls?

A question from Gelman - Regression & Other Stories... In answer to my own question...my understanding is that a difference in means should not capture the treatment effect when there are pre-...
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Bayesian model comparison with systematic error

Two parameters $(x,y)$ were measured for 3 different objects, wielding the following results: $$\{ (x,y) \}= \{ (1,3), (3,5), (5,9) \}$$ Knowing that the error in the estimation of the values $y$ is ...
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question of a Poisson random effect model

Assume a Poisson random effect model (with a random intercept) $$\log E(Y_{it}|b_i,x_i)=\beta_0+\beta_1x_i+b_i$$ $$b_i \sim N(0,\tau^2)$$ How to derive the population-averaged effect of one unit ...
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football probability

I have the following problem: Suppose that the probability of a football team (T) to win the match with 2-1 is p1 and, given that T scores a goal, the probability that the best player of the team ...
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find a suitable statistical method to analyze Yes/ No type qualitative data

I have done a survey on finding the applicability of Co- management system on a reservoir. This survey contains 8 main points. Under one main point, I included some Yes/No questions (No of questions ...
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Why does my ARIMA predictions on monthly data form a straight line?

For short detail, the goal was to forecast using 51 monthly observations of KPI of project implementations which I aggregated by sum from 463 observations from about 4 years of data (May 2017 to July ...
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Confidence interval for the parameter of a random variable

I am studying statistics, and I came across a problem that I am not sure if what I am doing is correct or if there is some other way to do this. Any feedback is appreciated. I divided in 2 problems, ...
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How to add change (in unemployment rate) to dependent variable

I'm an Econometrics beginner and I'm sorry if the title is confusing. I basically have a variable “unemp” which gives me the unemployment, the dependent variable of this model should be the change in ...
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Tail-equivalence implying same domain of attraction

Suppose two distributions F and G that have the same extreme point ($x^F = x^G$) and $$\lim_{x \to x^F}\frac{\bar{F}(x)}{\bar{G}(x)} = c \in (0, \infty)$$ Show that F and G belongs to the same domain ...
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How to test for significance with t-test and interaction variables? [duplicate]

I need to answer this question for my Econometrics class: "Does the gender gap differ significantly in urban versus non-urban areas? Test at the 5% level". I think I need to use the t-test ...
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computing the expectation of a variable

I'm trying to run a simulation but before I need to compute the exact value of X, where: V ~ Bernoulli(0.5) X|V ~ Normal(1-2 V, 1) The question is : what is the expectation of X (E(X)) in the ...
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Standardization in Machine Learning Models

Please answer this question in two contexts: Context 1 - Performance: Which models are sensitive AND which models are insensitive to standardization? Why? Are there any edge-cases in which ...
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What's wrong with this conditional probability working?

A past exam paper for my course (BSc Mathematics, second-year module in statistical inference and modelling, unpublished) has a question, Let $(X,Y)^T$ be a bivariate random variable with joint ...
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How to determine uncertainty of data from a bayesian posterior distribution

I am a bit confused as to how we determine the uncertainty of a set of data from Bayesian Analysis. In my specific case, I am asked the following: Assume $f(x,x_0)$ as the correct model for the ...
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Bayesian update with the shifted and scaled data

Suppose I have data $y$ (N observations) which follows a normal distribution: $y \sim N(\alpha+\beta*\mu,\sigma^2)$ while $\alpha$ and $\beta$ are known parameters. I want to update $\mu$ and the ...
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determining the sample size for quality test

It has been a long time since I did statistics in uni and I have a "simple problem" that I need solving. I need to determin for a sample size for a limited population of documents ( lets say ...
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5 votes
2 answers
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Antithetic method for monte carlo when bounds of the integral are infinite

I wanted to apply Monte Carlo with antithetic variables to estimate $\int_{0}^{\infty} e^{-x} \,dx$ (equal to 1). I used this R code. ...
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variance of Y based on a simple linear regression of X in which the slope and intercept are not constants

Consider a linear regression $y = aX + b$, where mean(a) = 5, SE(a) = 0.5; and mean(b) = 3, SE(b) = 0.1. When a and b are constants, $Var(Y) = a^2 Var(X)$. Do the SE's make a and b non-constants? If ...
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Why is SATE different to Treatment effect (difference in means)?

A question from Gelman - Regression & Other Stories. This one has me a bit stumped, I've reread sections of the chapter and I'm still not understanding why this is the case. I think the answer is ...
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1 vote
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Forming triangles from 4n points on a square [closed]

Suppose that n points are given at any side of a square. None of these points coincides with a vertex of the square. How many triangles are determined by these 4n points? The answer given is like so: &...
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Determining distribution based on first three even moments

Given that $E(X^2) = 1, E(X^4) = 3, E(X^6) = 9$, determine all possible distributions of X. I'm sure there's some insight I'm missing here. I noted that X is mesokurtic, so it's possibly normal... but ...
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Expected Prediction Error for 0-1 Loss Function

In ESL on pages 20 and 21, we have a derivation of expected prediction error of a classification rule $\hat{G}(X)$: $$ EPE(\hat{G}) = E_X\sum_{k=1}^{K}L[\mathcal{G}_k, \hat{G}(X)]P(\mathcal{G}_k|X) $$ ...
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Why is it called a ratio scale?

Can you provide a simple example that clearly explains- why ratio scale is called ratio scale?
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How do I prove the general formulation of LRV to verify that LRV for AR(1) is in fact (σ²)/(1-α)²

I understand the equations separately but don’t know how they are connected! Please give me an explanation / hint Given the AR(1) structure, the long run variance (LRV) of Xt is known to be (σ²)/(1-α)...
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How can I test for the unknown parameter of this model?

An asset price $p_t$ at any time $t = 1,2, \ldots, n$ is given by $x_t = x_{t-1}e^{\mu + \epsilon_t}$, where $\epsilon \stackrel{iid}{\sim} N(0,1)$. I was supposed to test $H_0: \mu = 0$ against $H_1: ...
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How do I do discriminant validity

How do I do discriminant validity the same as in this paper Page #333 "A Technological Pedagogical Content Knowledge (TPACK) Scale for Geography Teachers in Senior High School" https://www....
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2 votes
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Phase and amplitude in a periodic process

I am currently working out of Time Series Analysis and its Applications (R. H. Shumway & D. S. Stoffer 2ed) and am stuck of something that is said. The books says that if we have some periodic ...
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Mean term in simple linear regression

I am trying to derive the expression for the $E(y_i \epsilon_i)$ in simple linear regression. I substitute using $Cov(X,Y) = E(XY) - E(X)E(Y)$, so $E(y_i \epsilon_i) = Cov(y_i , \epsilon_i)- (E(y_i)...
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Obtaining a tractable expression for P(X<2Y)

The joint probability of the bivariate random vector $(X,Y)$ is given by $f_{X, Y}(x,y) = \frac{1}{2\pi}e^{-\sqrt{x^2+y^2}}$. Compute $P(X<2Y)$. My attempt: I made the obvious transformation to ...
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1 vote
1 answer
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Obtaining formulae for Poisson confidence interval

I have a sample $X=(X_1,\dots,X_n)$ of $i.i.d$ Poisson variables such that $n=100,\overline{X}=8.8$. My goal is to obtain a $80\%$ confidence interval for the parameter $\lambda=\theta$. That is, the ...
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1 vote
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P(XY<1/2) , X and Y uniform independent RV [-1,1]

I think the probability should be equal to shaded area shown in the figure divided by 4, which is equal to (3+ln 2)/4. Is this correct, because my answer is not matching My solution: I find the area ...
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4 votes
2 answers
99 views

Regressing on a variable, and then regressing the residuals on another variable

Consider observations on three variables $X1, X2,$ and $X3$. Suppose that $X1$ is regressed on $X2$. When the residual of the above regression is regressed on $X3$, the regression coefficient of $X3$ ...
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4 votes
3 answers
132 views

What is the chance all choices are different?

Suppose everyone (7 people) chooses independently and randomly out of 7 choices. What are the odds that no two people have picked the same option? I feel like the answer is $5040/823543,$ but I am ...
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For $Y \geq 0$, prove that $Pr(Y \geq k) \leq E(Y)/k$

Let $Y$ be a non-negative random variable, $k$ be any positive constant, show that $Pr(Y \geq k) \leq E(Y)/k$. My attempt (using integration by parts): \begin{align} \int_0^k y \,dF(y) &\leq E(Y) \...
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Using the Yule-Walker equation to calibrate an autoregressive model with the method of moments

Consider the following discrete autoregressive $\epsilon_t$, where $\epsilon_t \in (\pm 1 ) \forall \ t \geq 1$. We think of $\epsilon_t$ as the child of a previous sign at time $t-l$, where $l$ is ...
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true or false (p<0.05)

Can someone with more experience explain this statement to me? If one, with a hypothesis test, tests for equal means in two samples and gets the result p<0.05, does this imply that the ...
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Several p-values and understanding real relationships

I am trying to solve following exercise: Suppose I have 500 chemicals for affecting the bacteria growth. My null hypothesis is that applying chemical or not doesn't have any effect on bacteria growth. ...
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2 votes
1 answer
81 views

Probability that seems to exceed 1

Suppose we have the following information based on independent events: The probability of contracting Alpha variant is .4 The Delta variant is 50% more contagious than Alpha The Omicron BA1 variant ...
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1 vote
3 answers
59 views

What statistic test to use?

I am new to statistics! I have 2 groups of city population based on education level (group A: educated, group B: without education) I try to figure out if there is a difference between the number of ...
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1 vote
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Bootstrap Hypothesis testing: Why Null and alternative hypothesis are not mutually exclusive?

I'm new to hypothesis testing, and following an online tutorial which confuses me in the way they setup the null and alternative hypothesis, and compute the p-value. My understanding is that null and ...
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verify minimizing the summation of the Kullback–Leibler is equal to maximum likelihood estimate for theta

I wondered it seems like the maximum likelihood equals minimizing $\sum_x P(x)\log{P(x)}-log{\hat P(x)}$,as P(X) is the truth but why it equals to $\sum_x P(x)\log{P(x)}-P(x)log{\hat P(x)}$?I cannot ...
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3 votes
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A faster way of finding unbiased estimators for this linear model

No access to computers or calculators is available for this problem. Consider the following linear model $$Y_1 =\theta_1 + \theta_2 + \theta_3 + \theta_4 + \theta_5 + \theta_6 + \epsilon_1\\ Y_2 =\...
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$\chi^2$ test to determine whether different observations come from the same quantity

Three experiments are performed to measure the same quantity $T$. The three results are $$ \begin{cases} T_A = 990\pm 3\\ T_B = 920\pm 30\\ T_C = 950\pm 12 \end{cases} $$ Using a $\chi^2$ test, and ...
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3 votes
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$Y_1, Y_2, ... $ are iid Poisson($\lambda$)

We observe $Y_1, Y_2, ..., Y_T$ such that $T$ is the first $t\geq 1$ for which $Y_t>0$. Define $Y=Y_T$ (a) Find MLE $\hat{\lambda}$. (b) What is the relative bias, $[E(\hat{\lambda})- \lambda]/\...
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How is residual sum of squares derived from a given sample correlation coefficient and sample variance?

How would one calculate the residual sum of squares when given the other values (r, n, sample means of x and y, and both sample variances), but not the dataset? x is the predictor, y is the response. ...
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2 votes
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Deriving marginal distribution from a funky joint exponential-like distribution

Let $X$ and $Y$ be two random variables such that $$P(X > x, Y > y) = \exp\lbrack-(\lambda_1 x + \lambda_2 y + \lambda_{12} \max(x, y))\rbrack$$ What is the marginal distribution of $X?$ Source ...
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What is a model suitable for prediction of binary values based on time series?

There is a time series of stock quotes, length $n=500$. The data looks like: ...
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Overfitting the (non-nested) cross validation set

This is a follow up query to this query In one line: I wish to understand why is it that we will severely over fit the cross validation set (and hence need nested cross validation to correctly account ...
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