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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40 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 ...
1
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1answer
25 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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0answers
5 views

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 ...
1
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1answer
32 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 ...
0
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1answer
29 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 ...
0
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1answer
23 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 ...
0
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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
9 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 ...
0
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0answers
17 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 ...
0
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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?) . ...
0
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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 ...
0
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0answers
26 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 ...
0
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1answer
36 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
26 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
19 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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0answers
17 views

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 ...
0
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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 ...
0
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1answer
33 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!
0
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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: $$ ...
0
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1answer
15 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
25 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. ...
0
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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 ...
2
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2answers
30 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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0answers
55 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 ...
1
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1answer
36 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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0answers
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 ...
0
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1answer
17 views

$\def\Var{\operatorname{Var}}\Var \left[\frac{(n-1)S^2}{\sigma^2} \right] = 2(n-1) \Longrightarrow \Var(S^2) = \frac{2\sigma^4}{(n-1)}$

$\def\Var{\operatorname{Var}}$ $$\Var \left[\frac{(n-1)S^2}{\sigma^2} \right] = 2(n-1) \Longrightarrow \Var(S^2) = \frac{2\sigma^4}{(n-1)}$$ I know that when you take a constant out of the variance, ...
1
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1answer
30 views

Why is the sample Mean a consistent Estimator for the Logistic Distribution?

I think is this a very trivial question, but non the less: How can I show that the $ \hat\theta_n = $ $ \bar x $ is a consistent estimator of $ \theta _0 $. Since $ \theta _o $ is $ \mu $ for the ...
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0answers
16 views

which estimation are most effective(MY ATTEMPT)

To start with we have 2 variables $X_1$~$Bin(n,p_1)$ and $X_2$~$Bin(n,p_2)$. For example , we assume that we have an estimation $$p^*=p_1p_2(1-p_1p_2)$$ and another estimation ...
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1answer
42 views

Why is $\frac{\sum_{i=1}^n \log(X_i)}{n} = \overline{log X}$ [closed]

Why is $$\frac{\sum_{i=1}^n \log(X_i)}{n} = \overline{log X}$$ ($X_i$ are i.i.d samples)
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1answer
16 views

variance of the estimation $p^*=(x_1*x_2)/(n*n)$

I have that $X_1$~$Bin(n,p_1)$ and $X_2$~$Bin(n,p_2)$ and want to calculate the variance of the estimation $p^*=(x_1*x_2)/(n*n)$, which is an observation of the estimator ...
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0answers
18 views

Affine hypothesis

I'm looking at a data set containing income and expenditure on food for 235 household. We are interested in whether the cost of food depends on household income. I have verified that a workable model ...
0
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1answer
37 views

Why does $\hat\sigma^2 = \frac{1}{n}\sum_{i=1}^{n}X_i^2-(\bar X)^2= \frac{1}{n}\sum_{i=1}^{n}(X_i^2-\bar X)^2 = \frac{n-1}{n}s^2$? [closed]

Why does $$\hat\sigma^2 = \frac{1}{n}\sum_{i=1}^{n}X_i^2-(\bar X)^2= \frac{1}{n}\sum_{i=1}^{n}(X_i-\bar X)^2 = \frac{n-1}{n}s^2$$?
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2answers
103 views

domino's pizza claim

I just got a dominos promotional flier through the post and one of the graphics advertising 'create your own pizza' lists the various toppings and claims there are 'more combinations than people in ...
0
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1answer
23 views

Calculating the current age pdf from the lifetime pdf

Let's say I know the form of the lifetime pdf for some object class. If I select an arbitrary object from the class which is still alive and for which I have no ancillary information on its current ...
2
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0answers
41 views

Uniform Boundedness: Am I right or my TA?

I am a student, and I disagree with the solutions our TA has prepared. I am seeking verification that I am correct or explanation as to why I am wrong. It seems to be a disagreement or ...
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1answer
19 views

limiting variances of iid sample mean

In the book Statistical Inference (George Casella 2nd ed.), page 470, there is an example: $\bar{X}_n$ is the mean of $n$ iid observations, and E$X=\mu$, $\operatorname{Var}X=\sigma^2$. "If we take ...
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0answers
19 views

Do higher order sample moments converge to the distributional mean?

The Methods of moments estimation is based on the law of large numbers, which says that the sample means of i.i.d. random variables from any distribution converge to the distributional mean as the ...
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11 views

Confidence interval using weigthed mean.

I'm doing an experiment in which I take n measures and their respective weight. Mean and variance population are unknown. If I don't care about their weights, I can do confidence interval using a ...
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1answer
49 views

Show the consistency of an estimator?

Let $Y_1,Y_2,Y_3,...,Y_n$ be a random sample from the exponential distribution having PDF $f(y;\lambda)= \lambda e^{-y\lambda},$ $y>0.$ A) Show that $\hat\lambda_n = Y_1$ is not consistent for ...
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0answers
36 views

Hazard function for proportional odds model

The Cox proportional hazards model for survival data with covariate ${\bf z}$ is defined through the hazard function $h(t,{\bf z})$ by $$ h(t,{\bf z}) = h_0(t)~\cdot\theta~~,~~~\theta = \theta(\beta, ...
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1answer
18 views

What is the difference between $\sigma, \sigma_{\bar{x}}, S, s,$ and $s_{\bar{x}}$?

What is the difference between $\sigma, \sigma_{\bar{x}}, S, s,$ and $s_{\bar{x}}$? My textbook uses lots of different symbols, and it's not clear to me what the difference between all of them are. ...
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0answers
22 views

Variance of log-odds ratio

For a 2x2 table: I know that $\widehat{var}\left(log\left(\widehat{OR}\right)\right)=1/a + 1/b + 1/c + 1/d$. I'm trying to use a Taylor Series approximation to show this, but I'm getting a bit ...
0
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0answers
28 views

A strange derivative. Why it is equal to the expectation of ${\bf{r}}({\bf{x}},{\bf{z}})$?

Suppose probability distribution of $\bf{x}$,$\bf{z}$ are defined as $P({\bf{x}},{\bf{z}}|{\bf{\theta }}) = \frac{{\exp \{ {\bf{r}}({\bf{x}},{\bf{z}}) \cdot {\bf{\theta }} + {r_0}({\bf{x}},{\bf{z}})\} ...
0
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1answer
15 views

Probability of taking a random sample of 24 measurements and getting a mean of at least 103.6 of true population

A random sample of size n = 24 measurements is drawn from a normal population. The sample has a mean of 103.6 and a standard deviation of 12.5. If the true population is 100, find the probability of ...
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0answers
19 views

Questions on the theory of Lasso

The linear model ${\bf Y}={X\beta}+\epsilon$, where ${\bf Y}$ is a $n\times 1$ vector, and ${\bf X}$ is $n\times p$ matrix. $n\lt p$ and $rank({\bf X})=n$. $\epsilon\sim N(0, \sigma^2)$. How to prove ...
2
votes
1answer
39 views

Why isn' t high order polynomial a good fit?

Let's say I have a set of data points $(x_i,y_i), i=1,2,...,N$, and I want to approximate it using a polynomial $p(x)=\sum_{i=0}^n a_i x^i$ with a least squares fit (so $n<N$). I know that the ...