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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.

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1answer
59 views

Convergence of $U_n=\frac{1}{\sqrt{2n\sigma^2}}\left(\Sigma X_j-\Sigma Y_j\right)$ - central limit theorem

Suppose that $U_n=\frac{1}{\sqrt{2n\sigma^2}}\left(\Sigma X_j-\Sigma Y_j\right)$, where $X_1,X_2,\ldots$ and $Y_i,Y_2, \ldots$ are i.i.d. sequences of random variables with mean $\mu$ and variance $\...
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2answers
48 views

Prove convergence in distribution for n times the minimum of an unknown positive distribution

Let $Z_1, Z_2, ...$ be independent and identically distributed random variables with some density $f$. Suppose that $P(Z_i > 0) = 1$, and that $$ \lambda = \lim_{x\to 0} f(x) > 0$$ Let $X_n = ...
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0answers
12 views

Prove bi-directional relationship between convergence in distribution and convergence of probability mass functions

Let $X$ be a random variable that is positive and integer-valued. Let $X_1, X_2, ...$ also be random variables that are positive and integer-valued. Prove that $X_n$ converges in distribution to $X$ ...
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1answer
41 views

Finding joint support of $(XY,X/Y)$ where $(X,Y)$ has joint pdf $1/x^2y^2$ for $x,y\ge1$

$X$ and $Y$ are random variables with joint pdf $$\frac{1}{x^2y^2}\qquad,\, x\ge1,y\ge1$$ Set $$U=XY, V=X/Y$$ Explain why the joint range of $U$ and $V$ is given by: $$\{(u,v):v\in(0,1),u\ge1/v\} \...
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2answers
159 views

Testing whether $X\sim\mathsf N(0,1)$ against the alternative that $f(x) =\frac{2}{\Gamma(1/4)}\text{exp}(−x^4)\text{ }I_{(-\infty,\infty)}(x)$

Consider the most powerful test of the null hypothesis that $X$ is a standard normal random variable against the alternative that $X$ is a random variable having pdf $$f(x) =\frac{2}{\Gamma(...
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0answers
24 views

Why divide by 1-leverage?

I'm reading about resampling methods, and specifically leave-one-out cross-validation. I understood the method, and how to calculate the estimate of the test MSE (Mean squared error): In the setup ...
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0answers
12 views

Which test to use from a paired data that has only 4 cases of information from each group

I have a question from an exam: A Statistics Professor believes that adding videos of statisticians consulting on applied problems to her standard instructional material will help students solve ...
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1answer
35 views

How to prove the concentration equality for standard normal?

The following inequality is given in some of Yale's online lecture notes $$P(|Z|>x) \leq 2 \sqrt{2 \pi} \phi(x)$$ Where $Z \sim N(0,1)$ with density $\phi(x)$. They call it a concentration ...
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0answers
19 views

Confused with What Kind of Test I Should be Using

I have a question from an exam: The chef rates the taste quality of 5 different brands of anchovies. The brands came packaged in both a jar and a can. The chef rates 10 combinations of brand/...
0
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1answer
31 views

How to get standard deviation when the only given information is sample mean and p-value

I have a question for an exam: You read a journal article on TOTAL-cholesterol, HDL-cholesterol and LDL-cholesterol from a study of n=100 patients where TOTAL-cholesterol = HDL + LDL cholesterol. ...
2
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2answers
51 views

Calculating a probability based on a joint distribution between a Uniform random variable nested within a Uniform(0,1) random variable

Let $X_1 \sim Uniform(0,1)$, and $X_2 \sim Uniform(0, x_1)$, where $x_1$ is the realized value of $X_1$. Find $P(X_1 + X_2 \geq 1)$. I know that I need the joint distribution of $X_1$ and $X_2$. $...
1
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1answer
60 views

Profit maximization problem using linear regression

I'm currently on a university assignment where I'm stuck more or less in the middle. I have to answer the following problem: Suppose you are interested in estimating the production function for ...
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0answers
25 views

A question about sample partial correlation coefficient and multiple correlation coefficient

Question: My problem: All possible formulae of Multiple and Partial Correlation Coefficient I know :- But I am not sure, which one to use to know about the situation or condition mentioned in the ...
1
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1answer
27 views

Bonferroni interval - quick example calculation

Problem statement: Calculate the Bonferroni interval for any pair of means $\bar{y}_{.j},\bar{y}_{.k}$ with $I$=10 groups with equal sample size 30 in each group. Write this as a multiple of $s$. ...
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0answers
39 views

A problem about Partial Correlation Coefficient

Question: Prove that for a set of p variables, if all the simple correlation coefficients are negative, then all the partial correlation coefficients of all orders are also negative. My problem: I ...
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0answers
82 views

Finding the UMP size $0.1$ test with $U\sim\text{Unif}(\theta, \theta+1)$

$U$ is a uniform $(\theta, \theta+1)$ random variable, where where $\theta\in(−0.5,0.5)$. Consider testing $$H_0:\theta\geq0$$ $$H_1: \theta \lt0$$ (a) Give the test function of a ...
1
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1answer
72 views

A question about Trivariate Normal Distribution

Question: Let $(X_1,X_2,X_3)\sim N_3\left[\mathbf0, \begin{pmatrix}1&\rho_{12}&\rho_{13}\\\rho_{12}&1&\rho_{23}\\\rho_{13}&\rho_{23}&1\end{pmatrix} \right]$ Show that ...
0
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1answer
25 views

Two- Sample T test on Employment Data

I'm trying to conduct a two sample t-test on this data on food and drink workers in the Seattle MSA to see if there is a mean difference in levels of employment before and after the enactment of an ...
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1answer
24 views

Prove convergence in distribution, probability, or quadratic mean for a sequence of binary variables that depend on another binary variable

Suppose that $X$ has the support set $\{1, -1\}$, and $P(X = 1) = P(X = -1) = 0.5$. Suppose that $X_n$ has the support set $\{X, e^n\}$, and $P(X_n = X) = 1 - \frac{1}{n}$ $P(X_n = e^n) = \frac{1}...
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2answers
82 views

Prove that the sum and the absolute difference of 2 Bernoulli(0.5) random variables are not independent

Let $X$ and $Y$ be independent $Bernoulli(0.5)$ random variables. Let $W = X + Y$ and $T = |X - Y|$. Show that $W$ and $T$ are not independent. I know that I have to show that $P(W, T)$ is not equal ...
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1answer
43 views

Type 1 and Type 2 errors trade-off

Reducing Type 1 error will always result in increasing the Type 2 error This statement is false. I understand the definitions of Type 1 and Type 2 errors. What I understand is that there is, in fact, ...
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0answers
12 views

Find joint distribution for two different cases Kruskal Wallis

I'm a bit stuck with my homework in a subject called "Non-parametric Statistics". The task is related to Kruskal-Wallis test. The task is as follows: Let's look at the comparison of 3 independent ...
0
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1answer
18 views

Hypothesis testing for difference in proportions when given only the 25th percentile (Q1), Median, and 75th percentile (Q3)

I have a question for an exam: A researcher hypothesizes that the proportion of students who spend more than $200 on books is greater for students in the social sciences than that for the students in ...
1
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1answer
92 views

For three variables $X,$ $Y,$ and $Z,$ prove that the sum of all the pairwise correlations is atleast $-3/2$

As I can understand about the question, it is required to prove, $$\operatorname{Corr}( X, Y ) + \operatorname{Corr}( X, Z ) + \operatorname{Corr}( Y, Z ) \geq -3/2 \tag i$$ But for any two ...
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1answer
27 views

How can policies be ordered in reinforcement learning?

Following Sutton, Barto "Reinforcement Learning: An Introduction", in 3.6 Optimal Policies and Optimal Value Functions they define an ordering between policies: A policy $\pi$ is defined to be ...
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0answers
9 views

Calculating Standard Error for Crude Rates vs Age-Adjusted Rates

I have learned that the standard error (SE) for a crude death rate is simply the crude death rate divided by the square root of the number of deaths. Why then isn't the formula for calculating the ...
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0answers
25 views

CLT approximation - how large should sample be so probability is equal to 0.95? [duplicate]

We have a measurement which has mean $\mu$ and variance $\sigma^2$ = 25. Let $\bar{X}$ be average of $\textit{n}$ such independent measurements. How large should $\textit{n}$ be in so that $P(|\bar{...
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13 views

A question about the change in value of multiple correlation coefficient on multiplying each value with a variable quantity

Question: I know the formula of multiple correlation coefficient is; ( |R| is the correlation matrix and R11 is the cofactor of the (1,1)th element of R. But I really cannot figure out, how to deal ...
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1answer
26 views

Matrix Approach to Linear Regression Model

How do we interpret a data matrix, $X$ which does not have $1s$ as the first column? Does it refer to No Intercept Form? Could it also be interpreted as the mean deviated form? I understand that for ...
2
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1answer
35 views

The probability distribution of waiting time until two exponentially distributed events with different parameters both occur

I am working on a problem related to the waiting time until a parking garage is empty. We are given that the cars independently spend an exponential distributed time in the parking garage, with ...
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0answers
47 views

To find the covariance given the joint probability density function.

Question: I was solving some question papers and got stuck in this problem. My problem: I know how to find "marginal probabilities" from a joint probability density function and also know how to ...
0
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1answer
40 views

Finding joint probability distributions from marginal distributions

Question: I was solving test papers where I found this one. My doubt: I know to work with conditional probabilities and Jaccobian Transformation and part A and B can be done applying the above..But ...
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0answers
152 views

Step-by-step tutorials on Bayesian Networks

I'm trying to study Bayesian Networks (BN), but during classes [1] we covered just some theory and no exercises were given. Therefore I'm looking for step-by-step tutorials or solved exercises for BN ...
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75 views

Multiple regression - Coefficients testing

I have a multiple regression model $Y=12.45+0.072X_1-0.15X_2+0.03X_3+0.17X_4$ with $R^2=0.972$ A second regression model is given as follows $Y=13.77+0.072X_1$ with $R^2=0.855$ The number of the ...
5
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1answer
56 views

Is it possible for a Bayes estimator to be independent of the sample?

I am trying to derive the Bayes estimator. Without getting into the nuts and bolts of the question, basically I have an indicator loss function of the form $$L(\delta , \theta) = \mathbb{1}\{A\}$$ ...
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0answers
141 views

Probability of sum of sequences of integers

Let K be a positive integer.Suppose that the integers 1,2,3,...,3k+1are written down in random order.What is the probability that at no time during this process, the sum of the integers that have been ...
3
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1answer
29 views

Deriving posteriors in a hierarchical model

So I'm doing a problem with the following set up: $$X | \mu_1 \sim N(\mu_1, \sigma_1^2)$$ $$\mu_1 | \mu_2 \sim N(\mu_2, \sigma_2^2)$$ $$...$$ $$\mu_{k-1} | \mu_k \sim N(\mu_k, \sigma_k^2)$$ where $\...
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0answers
38 views

Using the Pbinom function in r to solve the posterior probability of rolling a dice 6000 times

I'm a newbie to stats and have been working on a question for about an hour now. You are testing dice for a casino to make sure that sixes do not come up more frequently than expected. Because you ...
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0answers
20 views

Differntiate between “Sample Multiple Correlation Coefficient” and “Population Multiple Correlation Coefficient”

I know what is a "Multiple correlation Coefficient". If Multiple regression of x1 on x2, x3, ..., xp is linear and the regression equation is given by, x1= x_1.234...p + e Then the product moment ...
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0answers
53 views

To find the Most Powerful Test (MP test) of the given hypothesis problem

A friend of mine asked me the question below on testing: Let $X$ be a single observation from one or other member of the family $\{f_0(x),f_1(x)\}$ where $$f_0(x)=\frac{1}{2^{x+1}}\mathbf1_{x\...
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1answer
77 views

Limiting distribution of a ratio using Basu's theorem

Edit: there's seems to be a typo in original question. This is a past exam question that I'm trying to solve. Suppose that $X_1,\ldots, X_n$ are i.i.d. Uniform (0, $\theta$) random variables. Let $...
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0answers
62 views

MASS{lda} plot in R [closed]

I'm reading the Introduction to statistical learning with R currently, but I blocked through a Lab about Discriminant analysis. So the thing is that we trying to fit a linear discriminant analysis ...
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0answers
32 views

Pearson differential equation distribution: gamma distribution derivation

How do I do 30a? It becomes extremely complicated if I solve the diff eq. with all the constants involved...
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0answers
26 views

What are good online course/s for learning optimization from beginning to advanced (linear, convex, integer, geometric, nonlinear, ..etc)?

I need some online course or group of courses, preferably videos, that allow me to understand optimization from the beginning. Give me a solid base knowledge (what is linear programing, what is convex,...
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1answer
96 views

What should my critical region look like in this LR test for shifted exponential distribution with pdf $e^{-(x-\theta)}\mathbf1_{x>\theta}$?

I have a small confusion over describing the cutoff point for the critical region in a likelihood ratio test when the null hypothesis is composite. Take this exercise in particular: Let $(X_1,X_2,\...
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0answers
7 views

Mixture distribution and priors

I had a quick question - I'm trying to use Gibbs sampling to sample from a Poisson distribution. According to the question, I am told that we are 90% certain that the prior is an exponential ...
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0answers
17 views

What is the significance of conjugate joint prior for normal distribution?

For example, for a normal distribution N(μ,σ2), it is possible to place, say, a separate normal prior on μ and another separate IG prior on σ2, but what is the prrof for conjugate joint prior ...
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1answer
78 views

Proof of Neyman Pearson Lemma

I am trying to understand the proof of Neyman Pearson Lemma as Uniformly Most Powerful test from here (Page 3). It says the following: Let $H_0: \theta = \theta_0$ and $H_a: \theta = \theta_1$. ...
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0answers
59 views

Posterior mean estimator with MCMC (Metropolis Hastings Algorithm) - Concrete example

I have a little project for which I have to estimate parameters on a PSF (Point Spread Function = response of the system to a dirac, i.e a star in my case). I have the 6 parameters to estimate : $p=(\...
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1answer
44 views

How to find joint distribution of $\min(U_0,U_1)$ and $\min(U_1,U_2)$ where $(U_0,U_1,U_2)$ are i.i.d Uniform?

I have this homework question where there are 3 random variables $(U_0,U_1,U_2)$ which are independent and uniform in the interval $[-1,1]$. I have two other random variables $(X,Y)$ defined as ...