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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calculating the risk function $max(\bar{X},2)$ under Squared error loss function

suppose $X_1,X_2,\ldots,X_n$ be a random sample of $N(\theta,1), \theta>2$. how can I calculate the risk function $max(\bar{X},2)$ under Squared error loss function
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2answers
18 views

distribution of the indicator function of poisson

Let $X_1,\dots,X_n$ be independent random variables with poisson distribution Given indicator function $$ U_i=\left\{ \begin{aligned} 1 && X_1 \ = 0\\ 0 && X_1 >0 \end{aligned} ...
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13 views

showing $\bar{X}$ is inadmissible estimator of under Squared error loss function

suppose $X_1,X_2,\ldots,X_n$ be a random sample of $N(\theta,1), \theta>2$. how can I show $\bar{X}$ is inadmissible estimator of under Squared error loss function
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9 views

Some true/false statements about MLE and UMVUE for a normal distribution

Let $X_1,X_2,...,X_n$ (assume $n\geq 2$) be a random sample from an $N(\mu,\sigma^2)$ population where $-\infty<\mu <\infty$ and $\sigma^2>0$ are unknown. Which of the following statements ...
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1answer
9 views

To find the distribution of the random variable based on uniform distribution

Let $X_1,X_2,...,X_n$ be iid $U(-5,5)$ random variables. Then the distribution of the random variable $Y=-2\sum\limits_{i=1}^{10}\log(|X_i|/5)$ is (A) $\chi_{10}^2$ (B) $10\chi_{2}^2$ (C) ...
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1answer
29 views

Calculating 95% confidence interval for mean for a normal population

Consider a normal population with unknown $\mu$ and variance $\sigma^2=9$. To test $H_0:\mu=0$ against $H_1:\mu\neq 0$, a random sample of size 100 is taken. Based on this sample, the test of the form ...
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Mathematical Statistics (Confidence intervals) [closed]

I'm having trouble solving this problem from text.
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18 views

estimation of the parameters of generative process modelling second-price-auction

The generative process: There are 2 entities (A,B) entity A - is the exchange performing second-price-auction entity B - is somebody who is trying to understand the distribution-of-the-value people ...
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1answer
20 views

To calculate variance, given conditional distribution

Let Y be an exponential random variable with mean $\frac{1}{\theta}$, where $\theta>0$. The conditional distribution of X given Y has Poisson distribution with mean Y. Then, the variance of X is ...
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1answer
26 views

UMVUE for pdf $f_{\theta}(x) = \theta e^{-\theta x}, x>0$

Let $X_1,\ldots,X_n$ be a random sample from a pdf $f_{\theta}(x) = \begin{cases} \theta e^{-\theta x}, & x>0 \\ 0, & \text{otherwise} \end{cases}$, where $\theta>0$ is an unknown ...
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12 views

Fisher Expected Information for a Gaussian Process model

Suppose I have a two dimensional Gaussian process model (GP), defined by a squared exponential correlation function s.t: $$R(x_{i},x_{j}) = \exp\left(-\frac{|x_{i} - x_{j}|^2}{2}\right).$$ I am ...
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4 views

Most Powerful Test and Rejection Region of Gamma Distribution [migrated]

Let $X_1,...,X_n$ be a random sample from a Gamma $(\alpha,\beta)$ population, where $\beta>0$ is a known constant. The rejection region of the most powerful test for $H_0:\alpha=1$ against ...
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1answer
10 views

Calculating power of a Hypothesis Testing Problem based on Uniform distribution

Consider the problem of testing $H_0:a=0$ against $H_1:a=1/2$ based on a single observation X from U(a,a+1). The power of the test "Reject $H_0$ if $X>2/3$" is (A)1/6 (B)5/6 (C)1/3 (D)2/3 ...
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0answers
6 views

Standard Error in Sampling Distribution

I want to understand that why while calculating probability on a sample data we consider standard error not the standard deviation of the sample. I know that standard error is the variance of sample ...
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28 views

two-parameter exponential distribution

Now, I have some problems about the distribution of random variable. In my work, let $X_{i}$ for i= 1 to n be iid random variable from two-parameter exponential distribution. We known the Mgf of X is ...
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1answer
31 views

What does “except null set” mean?

I am watching a video on sufficient statistics here. In the video, the sufficient statistics is defined as follows. I am puzzled by the part in the red box. I know it is describing some trivial ...
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2answers
35 views

Confidence Intervals that Contain the Mean: Designing an Activity

This past Wednesday, I had my stat class do the following exercise: Roll a fair 6 sided dice 25 times. Take the sample mean of the face value. Using the standard deviation of the uniform ...
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1answer
20 views

A question in the proof of Rao-Blackwell theorem

This could be a naive question but I am just puzzled. I am learning the following proof of Rao-Blackwell theorem but get puzzled by the equation in the red box. Why the two expectations can be reduced ...
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22 views

calculate The maximum likelihood estimator of parameter $\mu$ according to $T$

suppose $X_1,X_2,\ldots,X_n$ be a random sample of $N(\mu,1)$. if $T=\sum_{i=1}^n I_{(X_i<0)}$ how can I calculate The maximum likelihood estimator of parameter $\mu$ according to $T$. ($\Phi$ is ...
2
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1answer
36 views

P-value - test at $5 \%$ if there is significant difference in fuel consumption between the two petrol grades?

A car owners want to investigate if gasoline consumption of his car depends on the fuel octane number. He therefore intend to "premium" and "regular" at random and computes each time the ...
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38 views

calculating $E(X_{(i)}| \sum_{i=1}^5 X_i)$

suppose 5.5,3.5, 2.5,4.5,2 be a random sample from of gamma distribution with parameters of $ \beta,\alpha=2$. if $Z_{(i)}$ be i-th order statistic a random sample of size 5 from $\Gamma(2,1)$, how ...
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1answer
23 views

(bookexercise) sign test of $H_0: \bar m = \bar 25$ against $H_1: \bar m < \bar 25$ there we have 15 known random observation

Follwing data desribes the measured fracture strength of 15 randomly selected units made by a new ceramic material $$20, 42, 18, 21, 22, 35, 19, 18, 16, 20, 21, 32, 22, 20, 24$$ At a previously ...
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basic experimental statistical analysis

I was looking through a friend's notes on computing uncertainties didn't quite understood the mathematical process. In order to compute velocity,v, a collection of time and distance through which ...
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1answer
30 views

determine the confidence level for the confidence interval $I_{\bar m}=(x_{(2)},x_{(9)})$

Let $x_1,...,x_{10}$ be a sample from a continuous random variable $X$ with the median $\bar m$. Determine the confidence level for the interval $I_{\bar m}=(x_{(2)},x_{(9)})$. $x_{i}$ is the ...
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1answer
28 views

Why the mean value of a Gaussian process is usually set to zero?

In most textbooks (e.g. Rasmussen's book on Gaussian Processes for Machine Learning) the mean value of a gaussian process is set to zero. Of course, this does not mean that all the values are expected ...
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1answer
22 views

Help with method(s) show an iterative method converges to a known fixed point

Are there any general techniques that can be used to show that an iterative method converges to a (known) fixed point?. In my current situation, I know the exact fixed point, but I am unaware of a ...
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1answer
33 views

Central limit theorem: What is the probability than more than 36 randomly chosen songs are required to fill a program which is 76 minutes long?

I am a little stumped by what this question is asking. A large playlist consists of songs with times which have mean 2 minutes ten seconds and standard deviation 15 seconds. What is the probability ...
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0answers
8 views

Finding a bayes estimator

Let $X_1,...,X_n|\eta~\exp(1,\eta)$ and $\eta$~$N(\mu,1)$, where $\mu\epsilon\Re$. Find the Bayes estimator $\eta$ under the squared error loss. After finding the joint likelihood of $exp(1,\eta)$ ...
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1answer
53 views

calculating $\mathbb E\left(\exp\left(\frac{1}{2}\sum_{i=1}^n X_i^2\right)\right)$ [closed]

suppose $X_1,X_2,\ldots,X_n \sim \mathcal N(0,\sigma^2)$. How can I calculate $$\mathbb E\left(\exp\left(\frac{1}{2}\sum_{i=1}^n X_i^2\right)\right)$$
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1answer
10 views

Confidence interval for Gamma parameter.

Suppose $X \sim G(1, \theta^{2})$ and $Y \sim G(2, \theta^{2})$. How would I go about finding the constant $k$ for which $$P_{\theta}\left(\theta \leq k \sqrt{X+Y} \right) = 0.95$$ I figure that $X+Y ...
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0answers
16 views

Estimation of systematic error $B=E(\mu^*(X))-\mu$

The sample of n=5 observations are normal distributed. Let B be the systematic error $$B=E(\mu^*(X))-\mu$$ what's the estimation of the systematic error? I really don't know how to get the ...
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0answers
15 views

calculating UMVUE of parameter $(1-\sigma^2)^-\frac{n}{2}$.

suppose $X_1,X_2,\ldots,X_n$ be random sample of $N(0,\sigma^2)$. how can I calculate UMVUE of parameter $(1-\sigma^2)^-\frac{n}{2}$. I know $T=\sum_{i=1}^n X_i^2$ is Sufficient and complete ...
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0answers
14 views

Does the scale have a systematic errors?

Let B be the systematic error as $$B=E(m^*)-m$$ there m is the correct value. to determine a scale has no systematic errors you use an object with a known weight 14.4(this is te correct value?) . ...
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1answer
24 views

How to compute “weight”?

Assume I have a list of 100 used cars, which the following information for each: year, make, model, mileage, selling price. How could I figure out how much does the year, make, model, and mileage ...
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1answer
14 views

How can I show $Y=\frac{\ln X_1}{\ln X_2}$ is an ancillary statistic?

Suppose $X_1,X_2$ be random sample with probability density function $f(x)=\alpha x^{\alpha-1}e^{{-x}^\alpha}$, $x>\alpha$, $\alpha>0$. How can I show $Y=\frac{\ln X_1}{\ln X_2}$ is an ...
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20 views

How can I find the rejection region of a test so it has significance $α$, if $T(X)$ is a sufficient statistic with a known distribution.

Suppose that $X_1, . . . , X_n$ form a random sample from a density function, $f (x|θ)$, for which $T$ is a sufficient statistic for $θ$. Define $H_0: θ = θ_0$ and $H_A: θ = θ_A$. If the distribution ...
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1answer
31 views

Hypothesis test: decide the test variable ic in normal distribution.

I know that the test variable is based on the estimation of the parameter. So, if we have an sample of size $n$ from $N(\mu,\sigma^2)$, $\sigma^2$ known, and want to test $H_0: \mu=\mu_0$, then the ...
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1answer
23 views

what does extreme really mean in P-value method

if $P:=P_{H_0}$(To get an equally extreme outcomes that the observed), what does "equally extreme outcomes mean then? For an example in this situation. How does he get that " When saying extreme then ...
2
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0answers
18 views

P-value hypthesis test(my bad - found same question in math stack) [duplicate]

textbook exercise How can I calculate P-value at a test of $$H_0: u=3.2$$ against $$H_1: u \neq 3.2$$ if we have the observations $2.0$, $3.2$, $3.8$, $2.5$, $3.3$, $2.8$, $3.0$ and $3.4$ which is a ...
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Give the appropriate type of rejection regions - my attempt and questions.

Let $x$ be an observation of $X$~$Bin(n,p)$. We want to test null hypothesis $H_0: p=p_0$. Give the appropriate type of rejection regions where a) $H_1: p<p_0$ (Could need some comment ...
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2answers
24 views

selecting Rejection Regions with two-sided alternative hypothesis

Let $x$ be an observation of $X$~$Bin(n,p)$. We want to test null hypothesis $H_0: p=p_0$. Give the appropriate type of rejection regions where a) $H_1: p<p_0$ (Could need some comment ...
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1answer
32 views

Find MLE and show that it is unbiased.

I'm trying to solve a problem but not sure how to approach it because of the weird density function: Would appreciate any constructive advice!
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1answer
21 views

Distribution of student $t$ ratio under the wrong mean

Suppose that we have an i.i.d. sample of size $n$: $X_1,\ldots,X_n\sim N(\mu_0,\sigma_0^2)$. Define: $$ ...
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1answer
14 views

Find the Fisher information matrix.

I'm trying to solve a problem: But the parameter does not include variance itself, but standard deviation... Not sure how to approach the problem in this case...
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2answers
24 views

Need Critique on my solution. Find confidence interval

I came across this problem in my self-study. I found a solution with the help from this forum (answer below) and posted my solution here. Assume that $X_i$, $i\in\mathbb N$ is a sequence of I.I.D. ...
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1answer
30 views

The distribution of roll of a dice $12$ times

What's the distribution of a variable $X$ if $X$ represents the number of times you get outcome $k$ when you roll a dice $12$ times? I thought that the distribution was a binomial distribution with ...
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2answers
28 views

confidence intervals for 20 different parameters - distribution, probabilit and most probable value.

I need help with the subexercise (c) in the following exercise. A researcher is planning a study where she must calculate confidence intervals for 20 different parameters. The intervals are ...
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54 views

How do I put together a set of modified conditional distribution into a single joint distribution?

I am abstracting my original problem to a simple scenario. Consider a bivariate multi-modal mixture of gaussian distribution, $P(x,y)$. When we slice through $x$ or $y$ we get a univariate multimodal ...
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1answer
34 views

How to figure out the respective sufficient statistic for a given vector of parameters?

Let $Y$ be a random sample from $N(\mu,\sigma^2)$ where both $\mu$ and $\sigma^2$ are unknown. Let $\theta$ be the vector of parameters of interest $\theta=(\mu,\sigma^2)$. I need to find the ...
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41 views

Limiting Distributions and the Weak Law of Large Numbers

I have that $Y_1, Y_2, ..., Y_n$ are i.i.d. Poisson random variables with mean 1, and that $U_n = \sqrt{\frac{\sum_{i=1}^{n}{Y_i^2}}{n}}$. Given that I have a sequence $U_1, U_2, ..., U_n$, I'm ...