The area of statistics that focuses on taking information from samples of a population, in order to derive information on the entire population.

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How to compare two tests according to the power of the test?

enter image description here Can the rejection region calculated from the problem? I'm a little bit confused by all this staff.
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t distribution : formula for the degrees of freedom

I understood why we are using a t distribution in this case , because the sample isn't big enough to approximate the true standard deviation of the population by the sample's . But what I can't find ...
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30 views

Finding the joint distribution and covariance matrix of a function.

Question: Let X and Y be two continuous random variables with joint probability density function $$f(x,y)=\begin{cases}\frac{1}{2} & \text{if} \ \lvert x \rvert + \lvert y \rvert \le 1 ...
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Casella and Berger Likelihood Ratio Tests statistic vs Wasserman LRT

It seems like there is a discrepancy between these two authors on what a LRT is. Casella and Berger state on pg. 375. That the LRT statistic is: ...
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9 views

Where can using two different test statistics in a hypothesis test lead you?

Do two test statistics at the same $\alpha$ value give the same TYPE 1 error rate and same decision? I think it is clear that they do give the same TYPE 1 error rate by definition, but do they always ...
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sum of dependent random variables: [on hold]

Suppose $h=x_1+x_2+..+x_n$ and we have pdf of each $x_i$ (means: $p(x_1 ),p(x_2 ),…,p(x_n )$ )and also $x_{(i )}$ ($i=1,..,n$) are dependent random variables: How can I find the pdf of h?
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What is the asymptotic value of the smoothed probability in a HMM model?

If I have a HMM model with a hidden markov chain $(S_t)_t$ with 3 states and if I assume that the distribution of the observation knowing in which state it is, is a normal. Do I know what is the value ...
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13 views

Help with Hidden Markov model and SMC methods

So its quite a long background i don't really know where to start but here goes. The background is as follows: Background Observation model As the target is moving, it measures the signal (RSSI) ...
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52 views

Proof: Pivotal Quantity

Can anyone give me a clue of how to address this theorem? Suppose that $T$ es a real-valued statistic. Suppose that $Q(t,\theta)$ es a monotone function of $t$ for each value of $\theta\in ...
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628 views

Kruskal Wallis - Effect size

I analyse 4 algorithms and 3 sets of metrics for each algorithm in which I apply the non-parametric Kruskal-Wallis test for each metric to detect any differences in performance between these ...
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25 views

How to calculate probabilities in a Bayesian network?

Consider the Bayesian network represented by the directed acyclic graph given below: We are given the following probabilities: P(tampering) = 0.02 P(fire) = 0.01 P(alarm | fire ∧tampering) = 0.5 ...
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18 views

Paired T-Tests vs Independent

The effectiveness of a training course is examined, and performance of each individual in a group is taken both before and after, and the differences are used in a paired T test. Would it be possible ...
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Testing the Uniformly Most Powerful Test against the alternative

Hi I am working on the following problem A single observation $X$ is made from one of three densities listed below with parameter space $\Theta=\{0,1,2\}$. \begin{align*} ...
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29 views

Expectation of the conditional density

What is the difference between E[$X_1$|$X_n$ = $x_n$] and E[$X_1$|$X_n$]? I have found the first one, by integrating x*$f_{X_{(1)}|X_{(n)} = x_{(n)}}$ (x). If anyone has pointers for finding ...
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25 views

Help with the Maximum Likelihood Estimator?

I'm really struggling to understand this and am trying to learn it for my upcoming exam. The question I'm trying to do is Write down the likelihood function and then find the Maximum likelihood ...
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Probability Help with finding mean and variance of estimators [closed]

Consider a random sample, $X_{1} , X_{2} , . . . , X_{n} (n > 2)$, from a distribution with mean μ and variance $σ^{2}$. You may assume that $σ^{2}$ is known. Three estimators are proposed for μ:$$ ...
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24 views

Finding Uniformly Most Powerful(UMP) tests of size $\alpha$

Hi I am working on the following problem: Let $X_1,X_2,\ldots,X_n$ be a random sample from a distribution with PDF given by ...
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Expectation Maximization (EM) for 3-dimensional parameter $(\alpha,\mu_2,{\sigma}^2)$.

Let $x_i$ where $i=1,...,100$ are iid observations from a mix of two normal distributions with means $\mu_1=0$ and $\mu_2$ and the same variance ${\sigma}^2$. If $\alpha$ is the proportion of the ...
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Can the original function be derived from its $k^{th}$ order Taylor polynomial?

Coming from a statistics background, I'll provide an example related to fitting a model to an analysis dataset. Let's suppose I suspect the relationship between the mean value of the outcome variable ...
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34 views

Divergence of Chi-squared statistic

I want to write "proof" that a ${\chi}^2$ statistic becomes larger and larger as the sample size increases. I have come up with the following: For ${\chi}^2=\sum_{j=1}^{n} \frac{(O_j - ...
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22 views

How to find sampling distribution S.D in this case

The distribution of the weights of 1000 students is normal with a mean of 55kg and a variance of 25. 100 random samples of size 16 are taken from this population. Determine the mean and standard ...
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1answer
38 views

Uniform Distribution Estimator (not MLE)

Does anyone know where the estimator $\hatθ = X _{(1)} + X_ {(n)}$ for a U(0, θ) distribution comes from? Where: $X _{(1)}$ = min$_i (X_i)$ $X_ {(n)}$ = max$_i (X_i)$ I know it is not the MLE, ...
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24 views

Expectation in Bayes rule

Let $L(\theta,\delta)=(\theta-\delta )^2e^{\frac{(\theta-100)^2}{900}}$ with $X\sim N(\theta,100)$ and $\theta\sim N(100,225)$ find the bayes rule. I already founded that posterior is ...
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Finding MLE and UMVUE for $\theta$ for the following distribution??

I was working on following problem: Let $X_1,X_2,...,X_n$ be a random sample from a distribution with PDF given by $$f(x|\theta)=\theta^{-c}cx^{c-1}e^{-(\frac{x}{\theta})^c}$$ a) Find the MLE for ...
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Finding minimal sufficient for the following distribution???

Hi I was working on finding the minimal sufficient for the following distribution $$f(x|\theta)=\theta^{-1}x^{\frac{1-\theta}{\theta}}I(0\le x\le 1),\,\,\,\,\,\,\theta>0$$ By factorization theorem ...
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16 views

Understanding the shape of T distributions

I'm trying to understand why a T distribution with a small sample size has fatter tails and what this means. My textbook says "...t distributions have more probability in the tails and less in the ...
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37 views

rigorous statistics book recommendations

I am learning statistical inference by myself, I have skim through a few books like Casella Hoggs and I find it omitted lots of details, for example, they didn't introduce the conditional expectation, ...
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22 views

Linear Regression with Paired Data

For a sample of paired data (x,y), t tests are performed for the slopes of the population regression lines of y on x and of x on y. The null hypothesis in both tests is $H0:β=0$. Is it possible for ...
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76 views

Expected value of an estimator: biased estimator?

Let $f_X(x_i)=\theta\cdot x_i$, $x_i\leq \sqrt{2/\theta}$ with $\theta=\frac{2}{x_{(n)}^2}$ (derived using the MLE-method). What is the expectation of estimator $\hat\theta$? I'd assume ...
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27 views

expected value of the average cubed

I can not resolve an issue of the book Mathematical Statistics of Shao, is as follows: If $E|X_{1}|^3$ is finite, get $E(\bar{X}^3)$ and $Cov(\bar{X},S^2)$ If $E|X_{1}|^4$ is finite, get ...
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28 views

Maximum Likelihood of single observation

I'm stumped on this problem... I keep getting an undefined answer of having to solve -20 = 0. The likelihood function I get is $e^{-20\alpha}$. So I have $y_i=$ $ \begin{cases} 1& ...
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Inferential Statistic -F-ration

If there is no treatment effect, we can expect the F-ratio to be (a) some value between +1 and -1 (b) 1.0 (c) some random valuable (d) 0 (e) cannot be determined
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hypothesis testing on exponential family

In class, the professor did present the next problem: Suppose you have a random sample $x_1,...,x_n$, but is unknown if the original distribution of the sample is gamma or exponential. You also have, ...
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29 views

Clarifying the assumptions about a paired t-test

I've wrote my question in red ink (see links). There are two questions that I have. Primarily I want to know why they concluded that "there is some evidence that there is some difference in mean ...
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Determine who is the best seller

The numbers below show the number of lollipops Betty and Sharon each month for a total of 12 months or a year. Using the data and plot below, can you determine who is the bestseller? Would it be ...
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Hypothesis test to justify a claim

So I have a question regarding hypothesis tests where i have to justify a claim with statistical evidence. It is as follows: The average number of accidents in previous years in a city has been 15 ...
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15 views

How to estimate the max of a population using the normal distribution equation on a small sample

I recently watched a documentary on Mathematics. In the show they managed to estimate the weight of the largest fish that the fisherman was likely to of ever caught in his career just by analysing one ...
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How to calculate survey bias due to preference for first answer?

I was recently given the results to a survey in which participants chose answers to questions they would be likely to randomly answer, and in which the survey population is known to have a preference ...
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878 views

Improper Uniform Prior Distribution

In Bayesian method, choosing the prior distribution is an important step when using the Bayesian method. When choosing prior, we consider the prior knowledge to choose which prior distribution is the ...
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59 views

Deriving Joint probability density functions

Question: Let X and Y be two continuous random variables with joint probability density function $$f(x,y)=\begin{cases}\frac{1}{2} & \text{if} \ \lvert x \rvert + \lvert y \rvert \le 1 & \\ 0 ...
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35 views

How to formulate and test this statistical hypothesis

first of all I'd like to clarify that my biggest problem with this topic is probably my inability to formulate it correctly, maybe the answer is found trivially on the internet but I'm not able to ...
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29 views

Testing statistic $\frac{MSS(X)}{MSS(Y)}$

Suppose a test statistic $\frac{MSS(X)}{MSS(Y)}$, where $MSS$ denotes Mean Sum of Squares, is to be used for testing the significance of the factor $X$. Do we need the assumption $$\mathbb ...
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Convergence of Sample Moments and Weak Law of Large Numbers

Given $\{X_i\}_{i=1}^n$, sequence of indepedent random variables such that $E[X_i]=\mu$, prove: $$a_r=\frac{\sum X_i^r}{n}\xrightarrow{p}E[X^r]$$ and prove: $$m_r=\frac{\sum ...
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38 views

An Estimator Based on Exponential RVs

Let $X_1$, $X_2$, $\cdots$, $X_n$ be $n$ random variables independently sampled from the exponential distribution $\text{exp}(1)$. Suppose $k \leq n$, and $X_{(k)}$ is the $k$-th order statistic, ...
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Relation of sufficent statistic to random mechanism in constuction of a randomized estimator?

After reading a about randomized estimators and sufficent statistics, the question weather the random mechanism is determined uniquely by the statistic struck me, but I cant seem to get an answer to ...
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Help: SPSS and Data Interpretation of Voters. Republican vs. Democrats (1993 election)(Almost finished)

Hello everyone, I am Julieta this time I get stuck in the following exercise. It is a statistical analysis of pools, the statement is quite long I will try to keep it short and put some links. Note: ...
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Hypothesis testing - comparing data with a function

I am a beginner in statistics. My problem is as follows: I have a set of $N$ observations $\{(x_i,y_i,e_i)\}$, where $e_i$ is the error in the $i^\mathrm{th}$ observation. The individual observations ...
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23 views

Interpter P-value. Is the following statement true or false, and where is the mistake?

I have the following question, Statement: A given exercise has the p-value of 0.08 and my alpha is 5%. The exercise was using a linear regression model to predict some future value ...
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Can the correlation of a random variable $X$ and $g(X)$ be $0$?

Question: Can the correlation of a random variable $X$ and $g(X)$ be $0$? My attempt: I don't believe it can because they are dependent by definition therefore $Cov(X,g(X)) \ne 0$ which means the ...
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$U$ as a random variable? And what's with this integration becoming $1$?

I'm confused about two things. For the set up, I am told that Likelihood function is $L(\theta)=\Pi f(y_i;\theta)$ for a distribution with pdf $f(y;\theta)$. Log likelihood function is ...