1
vote
1answer
25 views

generating random samples with a PDF

I have the PDF of a distribution from which it is not possible to get a closed from for the CDF or inverse CDF. Is there a technique that would allow me to generate samples from this distribution ...
0
votes
0answers
22 views

Unconditional sampling distribution of regression coefficients

I am trying to find the (unconditional) sampling distribution of a regression coefficient in a simple linear regression. The linear regression $Y = \beta_0 + X\beta_1 + \epsilon$ is conducted for $N$ ...
4
votes
2answers
77 views

To sample or not to sample?

I have the following issue (I want to predict sports results). Let $X$ be a discrete RV with $m$ possible outcomes, each having probability $p_i$, for $i=1,\ldots,m$. Assume that I have a large iid ...
0
votes
1answer
39 views

Sampling error with weighted mean

I am studying statistics and I am wondering when it comes to standard error or a sampling if the calculation changes when there are weights added. I have a weighted mean: $$\mu_{w} = ...
0
votes
2answers
34 views

Why is there a difference between a population variance and a sample variance

Sorry if this answer is simple but I was wondering why is there a difference between a population variance and a sample variance? I understand The variance is calculated as: $$\text{Var} = ...
-1
votes
1answer
20 views

Probability in a Random Sample [closed]

V. The mean monthly rental rate for a two-bedroom apartment in Atlanta is $\$982$ (Elle. September 1998). Assume that the population mean is $\mu = \$982$ and the population standard deviation is ...
0
votes
1answer
19 views

Consequences of violation of independence assumption in ANOVA test

Can you please explain me what is the consequences of violation of independence assumption in ANOVA test? I couldn't visualize it. Does Cochran Theorem relates to ...
0
votes
1answer
17 views

Confidence/Tolerance interval for a percentage of a population

I have a problem I'm not sure how to solve. It goes something like: ...
2
votes
1answer
92 views

Non-i.i.d Empirical Risk Minimization

I'm not a statisticians so please forgive me if I posed a silly question, but it's a real problem for me in my research. Suppose we have defined risk in a regression problem as $R(f)=\int l(f(x),y) ...
0
votes
0answers
15 views

Calculating person-time

Suppose that an investigator wants to measure the incidence rate of high blood pressure in the $1997-98$ freshman class of a university. Assume that there were $1000$ entering freshmen and that this ...
1
vote
0answers
23 views

Mean & SD of Sampling Distribution

A population consists of 4 numbers {0, 2, 4, 6}. Consider drawing a random sample of size n = 2 with replacement. (a) What is the sampling distribution of $\bar x$? Is this a normal distribution ? ...
0
votes
1answer
82 views

Deriving the variance of the sample mean $\mathrm{Var} (\bar{y})=\frac{1}{n}(1-\frac{n}{N})S^2$

For a population of size $N$ with a simple random sample size $n$ derive the formula $$\mathrm{Var}(\bar{y})=\frac{1}{n}\left(1-\frac{n}{N}\right)S^2$$ where $S^2$ is the population variance. Hint: ...
0
votes
1answer
21 views

Test for Validity of Artificial Samples

I have a model that actually is learned from some observed samples. Then I use the model to generate several artificial data. My question is: Which test should I use to test if the data is of the ...
0
votes
1answer
34 views

the random heights of north american women

The heights of North American women are normally distributed with a mean of 64 inches and a standard deviation of 2 inches. A random sample of four women is selected. What is the probability that the ...
0
votes
0answers
7 views

How can I calculate distribution of minima of sections of a continuous path (from a stochastic process)?

I have a long slab whose width is defined by a stochastic process, whose complete statistics I am aware of, say. I now cut it into smaller sections of uniform length, and calculate the minimum width ...
0
votes
0answers
26 views

Derivation of F distribution

Prove that the PDF of Snecdor's F distribution, given by: $$F=\frac{U/n_1}{V/n_2}$$ Where $U=\chi^2(n_1)$ and $V=\chi^2(n_2)$, is given by: ...
4
votes
2answers
203 views

Monte Carlo estimator of the number of 1's in a very long binary sequence

Preface The question below is related to a problem I am working on, which requires counting the number of times a logic-valued function evaluates "TRUE" given an input value. The size of my input set ...
1
vote
2answers
29 views

Statistics - Lost with this question

I'm having trouble doing this question because I don't know where to begin. Could someone walk me through this slowly so that I understand the thought process and how to approach questions like this? ...
0
votes
1answer
30 views

Estimate correlation coefficient of unknown variable

Consider variable y depends on variable x and z linearly. I have $100$ sample values of $y$ and corresponding $x$ but don't have any values of $z$. The functional model is $$y = \alpha_1x + \alpha_2z ...
0
votes
0answers
20 views

Statistics. Sampling data $\bar{X}$

I understand that a continuity correction is required when approximating a discrete random variable by the Normal distribution and usually $+$ or $- 0.5$ is added. However when sampling and using the ...
1
vote
2answers
35 views

Use z confidence interval to estimate population proportion

Which of the following must be true of a sample in order for it to be appropriate to use a $z$ confidence interval to estimate the population proportion? (A) The sample is a random sample from the ...
0
votes
0answers
52 views

Asymptotic/sampling distribution

Assume we have the following function: $$f(p) = \frac{1}{(1-p)d}\ln\left(\frac{1}{T}\sum_{t=1}^{T}\left[\frac{1+X_t}{1+Y_t} \right]^{1-p} \right)$$ where $d$ is a constant $T$ is a constant $X_t$ ...
1
vote
1answer
55 views

Is Sample Covariance Tied to a Specific Distribution

In many sources on data analysis, the author(s) talk about calculating covariance of the data, and the formula is given as such $$ \Sigma = cov(X) = E[(X-E[X])(X-E[X])^T]$$ This formulation is given ...
0
votes
0answers
13 views

Indepedence of sample means of two orthogonal Gaussian vectors?

Suppose $\boldsymbol{x}_{1}$ and $\boldsymbol{x}_{2}$ are Gaussian vectors with each distinct but arbitrary means and covariances, i.e., the elements of each vector are generally intra-correlated. ...
1
vote
0answers
61 views

Statistics - Uniform sample vs. Representative sample

I have a question concerning two different samples, with the first being more uniform that the second. a) Chance errors are likely to be smaller... using the first set of subjects using the second ...
0
votes
1answer
48 views

Sampling random numbers with a certain condition.

I want to randomly sample three variables that are conditioned by $$x_1 \le x_2 \le x_3$$ and $x_1\in [0,\, \ell]$, $x_2\in [0,\, \ell-\ell_1]$ and $x_3 \in [0, \,\ell-\ell_1-\ell_2]$. I have only ...
0
votes
0answers
22 views

propability of largest samples of repeated random divisions

I have a bunch of numbers $A_1=\{a_{1,1},\dots,a_{n,1}\in\mathbb R^+\}=\{1,\dots,1\}$, that get multiplied by independent uniformly $[0,1]$ distributed samples, e.g. $a_{i,2}=X_ia_{i,2}$. This process ...
1
vote
1answer
67 views

Uniformly distributed points over the surface of the standard simplex

I would like to generate points that are uniformly distributed over the SURFACE of a standard $k$-simplex ($k$ dimensions, $k+1$ vertices). One way to efficiently generate points that are uniformly ...
0
votes
0answers
47 views

Minimum sample size to obtain population mean?

Knowing the average of a population, how could I evaluate an expected sample average based on k_sample? Data Info: Variance and standard deviation: $\sigma^2= 0.18500975256337462$ ...
0
votes
0answers
18 views

Finding properties of Inter arrival times

I apologise in advance if this is a bad question or if it's in the wrong place. Say I get a sample of inter arrival times, I'd like to have some kind of a measure which would tell me whether the ...
0
votes
1answer
28 views

How can i simplify the following term to get the right side?

$$\sum_{h=1}^{L}\frac{W_h^2S_h^2}{n_h}=\frac{1}{n}\sum_{h=1}^{L}{(W_hS_h)}^2$$ where, $n_h=\frac{n}{\sum_{h=1}^{L}N_hS_h}N_hS_h$ $\quad\text{and}\quad$ $W_h=\frac{N_h}{N}$ $\quad\text{and}\quad$ ...
0
votes
0answers
129 views

Proportional random sampling from a set of variable combinations

I have this table of variables, they can have the following values: $v_1 \in \{1,2\}$, $v_2 \in \{1,2,3\}$, $v_3 \in \{1,2,3,4\}$, I need to generate all the possible combinations $v_1v_2v_3$ of ...
2
votes
0answers
105 views

Central Limit Theorem Clarification

The Central Limit Theorem states that the sampling distribution of the sample mean: Converges in distribution to a normal distribution. Has an expected value (mean of the sampling distribution of ...
1
vote
1answer
16 views

Deriving Statistical quantites from other quantities

Studying for a statistics exam and I am stumped on the following: For a set of sample data, the following values were found. Fill in missing values Variable: X N: ? Mean: ? SE Mean: 2.05 StDev = ...
2
votes
1answer
87 views

Sample size from population?

This is probably very rudimentary maths, but given a strict population size ($N = 20$ for example), is the sample size any number $<N$? For use in calculation of confidence intervals using a ...
0
votes
0answers
88 views

unbiased estimator of the area of the circle

the radius of a circle is measured with an error of measurement which is distributed normal with mean $0$ and variance $\sigma^2$,$\sigma^2$ unknown.Given $n$ independent measurements of the radius , ...
1
vote
1answer
89 views

Why Gibbs sampling needn't “remixing”

I am generating $\mathbf{x}^{(1)}, \mathbf{x}^{(2)}, \dots, \mathbf{x}^{(n)}$ using Gibbs sampling methods. So I want $\mathbf{x}^{(1)}, \mathbf{x}^{(2)}, \dots, \mathbf{x}^{(n)} \sim$ some ...
1
vote
0answers
29 views

Relation with $F$ distribution and $t$ distribution

If $X\sim F_{n,n}$ , then show that $$\frac{\sqrt n(\sqrt X-\frac{1}{\sqrt X})}{2}\sim t_n$$
0
votes
1answer
29 views

Equivalent standard error for different populations

So I have population A and population B. The demographics are similar (assumes that the true expected value is the same in both populations). Population B, however, is twice the size of population A, ...
0
votes
0answers
23 views

How $\sum_{h=1}^{L}\sum_{h=1}^{L}W_h^2S_h^2=(\sum_{h=1}^{L}W_hS_h)^2$?

Suppose a population is divided into $L$ strata. $N_h=$Total number of units in the $h^{th}$ stratum $N=\sum_{h=1}^{L}N_h$ $W_h=$stratum weight and $W_h=\frac{N_h}{N}$ $S_h^2=$stratum population ...
0
votes
0answers
84 views

Intuition of the Sample Skew and Kurtosis Distributions

What does the sampling distribution of skewness and kurtosis represent, if any? In particular, I bootstrapped some data (I assume that this computes sample statistic distributions) and found that ...
0
votes
1answer
76 views

Central limit theorem: question about √n and σ2

Still making (good) progress with my knowledge on statistic. Sorry if I ask lots of questions about this recently but I really like math.stachexchange and really appreciate the quality of the answers ...
1
vote
0answers
107 views

Interpreting the meaning of sampling distribution

I have asked a couple of questions related to statistics recently as I just started to study the topic again (I ignored my university course on statistics and I now eat my fingers in anger). I asked ...
1
vote
1answer
92 views

Required sample size to reduce the interval size by half

I have a small doubt regarding a statistics problem and would like a confirmation. If we want to reduce the length by half of the confidence interval (at 95%), we should: Multiply the size of the ...
0
votes
0answers
45 views

Stratified Sampling and the Central Limit Theorem

What can be said about the convergence rate of stratified sample means to a normal distribution, given different allocation schemes? Obviously, under very poor allocation, this convergence can fail ...
1
vote
0answers
49 views

Gibbs / MCMC sampling for sum of parameters - how to improve slow mixing?

Suppose I have a hierarchical Bayesian model, where my observational prediction, $y'$, is calculated as the sum of other parameters, ${\alpha_i}$. My observation equation (the likelihood) is: $P(y | ...
2
votes
1answer
249 views

How can I sample a bivariate Gaussian distribution using Gibbs sampling?

I'm trying to sample a bivariate Gaussian distribution using Gibbs sampling, but I think I don't have the correct conditional probabilities. According to this lecture slides, the conditional ...
0
votes
1answer
36 views

Survey vs interview size

I have a survey of size 120 and 20 interviews. Can I compare the results? Both groups answered same questions. Example: let's say a question was do you like to read novels ? Can I say that 20% of ...
0
votes
1answer
47 views

Sample size question / margin of error

I'm a complete novice in this area - so "explaining to me like I'm 5" would be most appreciated. Essentially, I've been tasked with changing the a relevancy algorithm for products displayed on a ...
2
votes
1answer
221 views

Proof that $\frac{(\bar X-\mu)}{\sigma}$ and $\sum_{i=1}^n\frac{(X_i-\bar X)^2}{\sigma^2}$ are independent

Let $X_i\sim N(\mu,\sigma^2)$ ; where$[i=1,2,\ldots,n]$ $Z_i\sim N(0,1)$ ; where$[i=1,2,\ldots,n]$ Proof that $\bar Z=\frac{(\bar X-\mu)}{\sigma}$ and $\sum_{i=1}^{n}(Z_i-\bar ...