0
votes
1answer
16 views

Does the parameter change during data generation in Bayesian Inference?

Let's assume that we have the following graphical model: This graph encodes the joint distribution $P(p,x_1,x_2,x_3,x_4) = P(p)\prod_{i=1}^{4}P(x_i|p)$. In the Bayesian inference, if we know ...
4
votes
1answer
121 views
+50

Unbiased asymptotic variance

Problem: Let $X_1,...,X_n$ be indep. r.v.'s that satisfy, for $i = 1,...,n$, $E(X_i) = \mu_i(\theta)$ & $\mathrm{Var}(X_i)= \sigma_i^2(\theta)$. $\theta$ is the parameter of interest and the ...
2
votes
0answers
36 views

Conditions on Poisson random variables to convergence in probability

Let $X_1,X_2,...$ denote iid random variables such that $X_j$ has a Poisson distribution with mean $\lambda t_j$ where $\lambda$ > 0 and $t_1, t_2,...$are known positive constants. a)Find conditions ...
1
vote
1answer
30 views

Computing P-value

In a book, from a sample they derived Mantel-Haenszel chi-square statistic $$\chi_1^2=1.41$$ And it is written that : this $\chi_1^2=1.41$ is associated with a one-sided P-value between $0.10$ and ...
0
votes
1answer
15 views

infer the initial state from draws

I went through binomial distribution and Chi-square test etc and got confused further. This question might be very basic and simple. I have three states (Combination of two colors, both has equal ...
0
votes
1answer
34 views

Mantel-Haenszel $\chi_1^2$ statistic

I was doing a particular example from the book Epidemiologic Research by Kleinbaum(example 15.6) and didn't understood some basic statistical aspect. ...
0
votes
1answer
37 views

Method of moments for Beta $(\alpha_1,\alpha_2)$ distribution

I am trying to solve for the first two moments of a Beta$(\alpha_1,\alpha_2)$ distribution. We know that the first moment is equal to: $\mu_1 = \frac{\alpha_1}{\alpha_1+\alpha_2}$ and the second ...
2
votes
4answers
129 views

Is it true that $\mathbb E[{\frac{X}{Y}]}={\frac{\mathbb E[X]}{\mathbb E[Y]}}$?

If $X$ and $Y$ are both random variables, does it hold $$\mathbb E\left[\frac{X}{Y}\right]={\frac{\mathbb E[X]}{\mathbb E[Y]}}$$ ??
1
vote
1answer
31 views

Difficulty in understanding statistics and inference

I have a random variable, X, with cdf $F$ and pdf $f$. I want to estimate a parameter of $F$, say mean, $\mu$. So what do I do? I construct an estimator, $Y_n$, with several random variable X1, ...
0
votes
1answer
68 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
39 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} = ...
0
votes
1answer
11 views

Weighted variance of a small sample

I am trying to calculate the variance of a small sample. I have the data: ...
0
votes
2answers
150 views

How to find the following integration

Let $X_1, \cdots, X_n$ be $iid$ normal random variables with unknown mean $\mu$ and known variance $\sigma^2$. How to find $E[\Phi(\bar X)]$, where $\bar X:=\frac{\sum_{i=1}^nX_i}{n}$, please? I guess ...
1
vote
0answers
24 views

A variant of Hoeffding's Inequallity

I'm new to concentration inequalities and I have a question related to Hoeffding's inequality. Let $X_1 ~ \dots X_n$ be a set of i.i.d random variables, s.t. $E[X_i] = \mu$, $Var[X_i] = \sigma^2$, ...
0
votes
1answer
23 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
30 views

P-P plot and Q-Q plot

How to draw P-P plot and Q-Q plot manually ? I have looked at different site and they explained in various way, such as one said for p-p plot in X-axis there is residual in ascending order and in ...
0
votes
1answer
29 views

Existence of Expectation

Question: Let $X_1$, $X_2$, and Z denote independent, real-valued random variables. Assume that $Pr(Z=0) = 1 - Pr(Z=1)$ = $\alpha$ for some $0 < \alpha < 1$. Define $Y = \left\{ ...
1
vote
1answer
16 views

Confidence interval for difference in means

I need to obtain a 95% confidence interval for the indifference in the mean score overall I have the following data which states subject (A-L) and then Test and Retest A B C D E F G H I J K ...
0
votes
1answer
19 views

Is the statement: $p\left(\left.y\right|h^{-1}\left(\varphi\right)\right)=p\left(\left.y\right|\varphi\right)$ correct?

Say I have a likelihood function $p\left(\left.y\right|\theta\right)$ and I make the reparameterization $\varphi=h\left(\theta\right)$ using the bijective function $h$ with inverse $h^{-1}$. Then it ...
1
vote
1answer
32 views

Doubts on Bayes hypothesis test

I meet one problem on hypothesis testing in statistic theory. "Assume given the probability spaces $(X,S,\mu_i)$, $i=1,2$. $H_i$, $i=1,2$, is the hypothesis that $T$ is from the statistical ...
2
votes
4answers
50 views

The Objectivity of Statistical Testing

I have a very generic question about applied statistics. Suppose, to make things simple, we have a biased coin with probability $p$ of landing heads. We want to determine if our coin is truly fair - ...
0
votes
2answers
38 views

probability question needs some help

suppose $X$ and $Y$ are independent and identically distributed random variables that are uniformly distributed on $[0,1]$ What is the PDF of $ W=Y-X $ i tried to draw a picture to illustrated it ...
1
vote
1answer
13 views

just a manipulation problem for statistics

I am a little bit confused when i was asked to Suppose $X_1, X_2,\ldots,X_n$ is a simple random sample from a continuous distribution with density function $f(x)$. Consider the new random variable ...
0
votes
0answers
34 views

Nonrejection region- equivalently the area determined by confidence interval for $H_0: \hat \beta=\beta^*$

For $H_0: \hat \beta=\beta^*$ I want to prove that the non-rejection region in level of significance approach Will be eqaul to the area determined by upper and lower bounds in confidence interval ...
1
vote
1answer
49 views

What is the pivotal quantity

I had a question of two parts. I solved the first part but I am stuck on the second. Any hints or partial solutions would be greatly appreciated. a)$ X_1,....X_n$ are uniform iid on the interval ...
0
votes
0answers
18 views

Calculate probability of total population from sample

I have a population of infinity size. Each Unit is Boolean(True/False). I take a random sample of size N of which M elements are True(and N-M are False). I'd like to know what the probability is that ...
0
votes
0answers
25 views

How to find MLE function?

I have two vectors of known values x and y. And the relationship between them is y=sin($\theta$$\cdot$x)+$\epsilon$, $\epsilon$~N(0,1) . The question is how i find the MLE function for $\theta$?
0
votes
0answers
18 views

Stationary point of Unnormalized and Normalized KL-divergence minimization

I have encountered a problem, basically related to the http://arxiv.org/abs/1206.6679 . If I want to minimize the normalized KL-divergence KL(Q||P) with Q a multivariate Gaussian distribution. but in ...
0
votes
1answer
43 views

Gaussian Approximation of an intractable distribution

I am currently encountering this problem: I have an intractable distribution and I want to minimize the KL divergence of this distribution and a multivariate gaussian distribution. So we just need ...
0
votes
2answers
71 views

What is this distribution???

Let $X_1, X_2, \ldots, X_n$ be a random sample from a population with $E(X_i) = \mu$ for all $i \in \{1,\ldots, n \}$. Define $ Y_i = \begin{cases} 1 & \mbox{ if } X_i < \mu \\ 0 ...
1
vote
1answer
55 views

asymptotic normality and central limit theorem

Here's the question Can somebody explain the difference between asymptotic normality and central limit theorem? They seem very similar to me.
0
votes
1answer
44 views

Fitting a function with relative uncertainties

I want to find the parameters of a function that better fit some measurements. Usually, I use least squares. With it I am assuming that the best function is the most likely one and that the residuals ...
4
votes
1answer
85 views

Philosophy of Statistics (Likelihood Function)

Last week during statistics class, my professor asked us a few basic questions about statistics. We could answer most of them except these three questions that we could not provide him good answers. ...
1
vote
0answers
18 views

Confidence bounds given random verification

[Edits made for clarification and brevity.] I'm working on an idea for a fault detection algorithm, and I've boiled it down (I think) to the following problem. A box contains 10 balls. The balls can ...
0
votes
2answers
46 views

How to find confidence interval of 0.95 in this problem?

For sample $x_1,\cdots,x_{100}$, following holds. $\sum_{k=1}^{100}x_k=400$ and $\sum_{k=1}^{100}x^2_k=2500$. Find the confidence interval of 0.95 for the population mean $m$. I've calculated the ...
0
votes
0answers
25 views

Computation of conditional probabilities

I am aware this question might not be well formulated, but it is not very clear for me neither, so if anyone could help me explicit it... I observe many examples e_i. For each example, I compute two ...
0
votes
1answer
127 views

Queuing Theory with Poisson Distribution

Suppose customers arrive in a one-server queue according to a Poisson distribution with rate lambda=1 (in hours). Suppose that the service times equal 1/4 hour, 1/2 hour, or one hour each with ...
0
votes
2answers
32 views

investigating a relationship at $5 \% $ level significance

$n=12$ $\bar{x}=? $ sample average = $\sigma $ standard deviation = $\alpha =$ $H_a :$ Degrees of freedom if applicable Critical value(s) = Sample mean Standard error of mean = $\frac{\sigma } ...
0
votes
1answer
28 views

Finding P value

I have these observations $(2,3.2,3.8,2.5,3.3,2.8,3.0,3.4)$ from $X \sim N(\mu,\sigma^2)$ and i want to calculate the $P$-value testing $H_0: \mu =3.2$ against $H_1 \neq 3.2$ with $\sigma = 0.6$ ...
1
vote
1answer
195 views

Finding MLE of $f(x;\theta) =1$ if $\theta-1/2<x< \theta+1/2$

Let $X_1,...,X_n$ have density: $f(x;\theta) = \begin{cases} 1 &\mbox{if } \theta-1/2<x< \theta+1/2 \\ 0 & otherwise \end{cases}$ Let $Y_1=min \lbrace X_1,...,X_n \rbrace$ and ...
0
votes
0answers
68 views

Find MLE of $\alpha$ of $f(x;\alpha)=(1+\alpha x) /2$ (stuck at derivative setup)

$X_1,...,X_n$ is an independent sample with common density: $f(x;\alpha)=(1+\alpha x) /2$ where $-1<x<1$ and $-1<\alpha <1$ I have to find the maximum likelihood estimate of $\alpha$. ...
0
votes
0answers
8 views

Statistic (Linear normal model): $X_{hi} \sim N(\alpha_h+ \beta t_{hi}, \sigma^2)$. How to calculate $C_{95}(\beta)$ and $t$-test for $\beta = 2.5$?

Statistic (Linear normal model): $X_{hi} \sim N(\alpha_h+ \beta t_{hi}, \sigma^2)$. How to calculate $C_{95}(\beta)$ and $t$-test for $\beta = 2.5$? I am in the statistical model $X_{hi} \sim ...
0
votes
1answer
22 views

Confidence intervals for mutliparameter fit

I am doing fits to some data and currently using mathematicas nonlinearmodel fit to generate fits and CIs, as well as doing bootstrapping for CIs for completeness. However, I would like to know how to ...
1
vote
1answer
35 views

Variance with correlated variables

A simple question that I don't manage to solve: I can use different methods to measure a magnitude $x$. The results of these methods are correlated and have some uncertainties. Combining the results ...
2
votes
3answers
109 views

Questions about Bayesian inference

From Wikipedia The prior distribution is the distribution of the parameter(s) before any data is observed, i.e. $p(\theta \mid \alpha )$. ... The sampling distribution is the distribution of ...
0
votes
0answers
20 views

Efficient algorithm for point estimation of a dependent random variable

Suppose $X$ is a normal-distributed random variable and $f$ is a known smooth function (possibly quite complicated, with many oscillations). Let $p(y)$ be the pdf of the dependent random variable $Y = ...
2
votes
2answers
43 views

Show that as $d$ goes to $\infty$, a standardized version of $X$ has the STD Normal Dist

I am currently stuck on this problem and I would greatly appreciate some help. The problem is as follows: Let $X$ have a chi-square with $d$ degrees of freedom. Show that a standardized version of ...
1
vote
0answers
27 views

Adjusting regression for small sample bias

I have a set of data points $\{x_i\}$. These data points are grouped so that (say) $i\in\{1,2,3\}$ is group $A$, $i\in\{4,5,6,7\}$ is group $B$, etc. I would like to test the null hypothesis of no ...
0
votes
2answers
59 views

$QQ$-plot - Why do we choose the empirical distribution $F_n(x) = \frac {\#\{y \in S \mid y \le x\}} n$, $S$ is sample, for comparison with normal?

$QQ$-plot - Why do we choose the empirical distribution $F_n(x) = \frac {\#\{y \in S \mid y \le x\}} n$, $S$ is sample, for comparison with normal ? Let $S$ be our sample of size $n$. Then we form ...
0
votes
2answers
29 views

integral support for a density function

$$f_{XYZ}(xyz)=ke^{-(x+y+z)}$$ $$ 0<x<y<z $$ I must find for which k this is a density function. $$\int_{Rx}\int_{Ry}\int_{Rz} ke^{-(x+y+z)} dzdydx =1$$ $$k\int_{Rx}\int_{Ry}\int_{Rz} ...