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Questions tagged [statistical-inference]

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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Log predictive density asmptotically in predictive information criteria for Bayesian models

I am reading this paper, Andrew Gelman's Understanding predictive information criteria for Bayesian models, and I will give a screenshot as below: Sorry for the long paragraph. The things that ...
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Let X1,…, Xn be i. i. d. with N(0,theta). Show that the summation from xi=1 until n from (Xi)^2 is a Sufficient statistics for theta.

Help me to solve this problem about sufficient statistics please.. Let X1,..., Xn be i. i. d. with N(0,theta). Show that the summation from xi=1 until n from (Xi)^2 is a Sufficient statistics for ...
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Let $X_1,\ldots,X_n$ be i.i.d from a distribution with pdf $f(x) = (1-\theta)^x\theta$; with $x=0,1,2,3,\ldots$ zero elsewhere.

Help me to solve this problem please. I am a new comer in math exchange. I have a homework from my lecture about sufficient statistics. Would you be so kind help me? Let $X_1,\ldots,X_n$ be i.i.d ...
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Variance of a sum of random variables with overlapping domains

This problem has stumped me for a while, so I'd love to get some fresh eyes on it. Problem Suppose an online retail provider served a large number of users ($N$) in a given time-window. Each user ...
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how to write the condition $\Lambda \beta$, in the linear model?

i'm dealing with the model: $$Y_{1j}=\mu+\tau_{1}+\varepsilon_{1j}, \hspace{2cm} j=1,2,...,n_{1}$$ $$Y_{2j}=\mu+\tau_{2}+\varepsilon_{2j}, \hspace{2cm} j=1,2,...,n_{2}$$ $$Y_{1j}=\mu+\tau_{3}+\...
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Likelihood function for continuos observations [on hold]

I need to know how to calculate the likelihood function for numeric range observations, for example: Given two observations of normal distribuition: $y_1 = a \leq y_1 \leq b$ and $y_2 = c \leq y_2 ...
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Derived parameter instead of parameter estimation

In a statistical model $$ (\mathcal X,\mathcal F, (P_{\theta})_{\theta\in\Theta}) $$ we call $$ \varphi(\theta)\in\mathbb R^d $$ a derived parameter (literal translation from German, Source: ...
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Let Xn be a sequence of random variables that converges in distribution to a random variable X. [on hold]

Let Yn be a sequence of random variables such that for any finite number c, limn→∞ P(Yn > c) = 1. Show that for any finite number c, ...
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Prove an equality consists of increasing events

If $A$ and $B$ are correlated increasing events with respect to $X$ as it shown below and $X$, $Y$ , $W$ and $Z$ be independent random variables, is it true that we say: $$P \left(A \cap B \right) \...
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Is it possible that probability of occuring intersection of two events will be greater than probability of occuring each of them?

If $A$ and $B$ are correlated events, Is it possible that we have : $$P\left(A \cap B \right) \geq P\left( A\right)$$ and $$P\left(A \cap B \right) \geq P\left( B\right)$$ Is it possible?
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Most powerful test for discrete uniform

Let $X$ be a random sample from a discrete distribution with the probability mass function $f(x, \theta) =\frac{1}{\theta} , x=1,2,...,\theta;= 0 \ \text{otherwise} $ where $\theta \ \text {is ...
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Show that E(Y | X) = E(E(Y | X, W) | X). The right hand side may sometimes also be written as E(E(Y | X, W) | X) = E(E(Y | W) | X).

We know that, E(E(X|Y)) = E(v(Y)) = E(E(X|Y=y)) = E(X), where v(Y) = E(X|Y=y). Hence, E(E(X|Y)) = E(X). Note: Let X and Y be discrete or jointly continuous random variables. The conditional ...
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Inference Statistic - Likelihood Function [on hold]

Can anyone help me understand this? Consider the four observations from de Normal Distribution with variance equal to one $y_1 < 10$$, y_2 > 10 $, $5 < y_3 < 10 $ and $ y_4 = 10$. ...
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1answer
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Limiting distribution of $T_n=\frac{\sum_{i=1}^n(2X_i−n)}{n}−3$ where $f_k(x) =\frac{x^k}{k!}e^{−x}I_{(0,\infty)}(x)$

$X_1$, $X_2$,..., are independent random variables. $X_k$ has pdf $$f_k(x) =\frac{x^k}{k!}e^{−x}I_{(0,\infty)}(x)$$ Let $$T_n=\frac{\sum_{i=1}^n(2X_i−n)}{n}−3$$ Consider the ...
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Range of Pearson's moment coefficient of skewness

The Pearson's moment coefficient of skewness is defined as $\gamma_1 = \operatorname{E}\left[\left(\frac{X-\mu}{\sigma}\right)^3 \right] = \frac{\mu_3}{\sigma^3} = \frac{\...
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2answers
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how to find the distribution of the least square estimator $\beta$?

i'm solving a problem that involve a linear model, and i'm trying to get the distribution of the least square estimator $\beta$. i found in a book that: $\widehat{\beta}\sim N_{p}(\beta, (X^{\...
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find a confidence interval in a linear model problem

i'm trying to solve a problem that involve a linear model given its normal equations, and the errors have a normal distribution but i'm a little lost. the problem is about construct a 95% ...
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Central limit theorem and confidence interval

I have some questions concerning the following problem especially about b). My problem is the following: Let $X$ be an observation from a $Bin(n,\frac{1}{2})$: a) prove that the ML estimator ...
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Does it make sense to use “linear transformation” to analyze effectiveness of a new technique?

I am relative new in statistical analysis, and I have question when I'm trying to measure the performance of new techniques. Imagine we have $m$ machines to improve with new technique, and $n$ tasks ...
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Finding umvue of normal pdf [closed]

Let $X_1,X_2,\ldots,X_n$ be iid $\mathcal N(\mu,1)$ RVs.Let $T(\mathbf X) =\sum_{i=1}^nX_i$.Show that $\phi(x;t/n,n-1/n)$ is UMVUE of $\phi(x;\mu,1)$ where $\phi(x;\mu,\sigma^2)$ is the PDF of a $\...
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What is an unconditional model for a time series variable?

If I am being asked to do an unconditional analysis of a time series variable, lets say GDP starting in 2000, what model am I supposed to estimate?
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Compare KS test and Wasserstein distance or Earth mover's distance.

Consider two sets of data points A and B. Both these data set are from mixture of unknown number of Gaussians. The mean of the Gaussians are little different for each set (there may have few overlap ...
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1answer
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Reasoning behind the statement 'Consistency is a property of a sequence of estimators rather than one point estimator"

I just started learning statistical inference course and I am doing this topic called consistency. I am not able to understand this line 'Consistency is a property of a sequence of estimators rather ...
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Understanding p-value and its relation with the value $\alpha$ used in tests

I am having a little trouble understanding these concepts. Let’s take a real problem, for instance. I buy some machinery, and I know newer models break down with an exponential probability of mean 5 ...
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1answer
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Which distribution to use

After many generations, it is known that 75% of students pass certain subject. if a random sample of 40 students is such that it could be consider they have the same characteristics as the ones ...
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1answer
35 views

Hypergeometric Distribution over an interval

In a village with 2000 people, 100 people suffer from Alzheimer's disease. On a certain day, 40 people are admitted to a hospital. Calculate the probability that between $15$ and $25$ people (...
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Binomial Distribution on finding the potency of a drug

An experiment is designed to test the potency of a drug on 40 rats. Previous animal studies have shown that a $10$-mg dose is lethal $10$% of the times within the first $4$ hours. What is the ...
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Linear Least square estimate of $x^3$ given $x$ and the moments.

I have been struggling to find a direction on how to proceed with the following problem. Given that $x$ is a zero mean (non-Gaussian) random variable with moments E$(x^n)=\mu_n$. I need to find the ...
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Statistical problem involving a coin

Two friends flip a coin $100$ times, and $58$ it is head. One says the coin is rigged, the other says it is not, and that it happened by chance. The problem asks to verify the hypothesis $p=1/2$ ...
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Student´s-T interpretation for problem

According to the history of certain shop department, the purchase exhibit in a ticket can be modeled by a random variable with mean \$1,025. For planning purposes, the new marketing director decides ...
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How do they simplify the autocorrelation function?

In my book they state that ...the autocorrelation function can be defined as $$r_X(s,t)=\frac{\text{Cov}[X(s),X(t)]}{\sqrt{\text{var}[X(s)]\text{var}[X(t)]}},\tag 1$$ where $X(t)$ and $...
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Let Y(1), Y(2), Y(3), Y(4), Y(5) denote the order statistics of a random sample of size 5 from a distribution [duplicate]

Let Y(1), Y(2), Y(3), Y(4), Y(5) denote the order statistics of a random sample of size 5 from a distribution having p.d.f. f(y) = e^(-y), 0 < y < ∞, zero elsewhere. Show that Z1 = Y(2) and Z2 = ...
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Let Y(1), Y(2), Y(3), Y(4), Y(5) denote the order statistics of a random sample of size 5 from a distribution having p.d.f.

Help me to solve this problem please.. Let $Y_{(1)}, Y_{(2)}, Y_{(3)}, Y_{(4)}, Y_{(5)}$ denote the order statistics of a random sample of size 5 from a distribution having p.d.f. $f(y) = e^{(-y)}, 0 ...
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1answer
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Sufficient Statistics: Proof of a lemma by Halmos and Savage

I am reading the paper "Application of the Radon-Nikodym Theorem to the Theory of Sufficient Statistics" by Halmos and Savage and have much trouble following the proof of Lemma 7. Below is (my ...
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Markov chain transition kernel inference

Say we consider a continuous markov chain $X$ with state space $S \subset \mathbb{N}^n$, transition kernel $P$, rate $\lambda$ and initial state uniformly chosen at random in $\mathbb{N}^n$. We do ...
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1answer
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Finding an unbiased estimator of$ e^{−2λ}$ for Poisson distribution

The random variable X has a Poisson distribution with unknown mean λ, where 0 < λ < ∞. Based on a single X , Is it possible to calculate an unbiased estimator of $ e^{−2λ}$ ? I have taken an ...
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How big of a sample size do you need to ensure the mean is within $\delta$ of the sample mean?

Visitors to a website click on an add with fixed unknown frequency $p$. You collect a sample of $n$ visits and compute the clickrate $\hat{p}$. Find the minimum $n$ such that $$P(|\hat{p}-p|<\delta)...
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1answer
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Calculating joint cdf from joint pdf

I have the following joint distribution function $f_{xy} = (12/5)(x+y^2)$ on $0<y<x<1$ (zero everywhere else) I know I must integrate along both $y$ and $x$ to find this: $$F(x,y) = \int \...
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Finding MLE when there are different distributions for different values of parameter

Suppose a random sample of size $n$ is drawn from a population having different distributions, one for different values of the parameter $\theta$. Let's say I consider the case when there are two ...
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Estimations from Standard Deviation

A while ago I was watching a show that centered around algorithms. At some point in it, the host of this show/documentary was given data from a fisherman about his haul of fish for that day. From this ...
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Student's-T distribution for problem

A biologist want to measure the mean height of a certain specie. If is assume $σ=2.5 inches$ and 100 males are randomly selected: a) Find the probability that the difference between the sample ...
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Bias of a function of the avarage

$\overline{x}$ is an unbiased estimator of the exact average. Now, we let's imagine that we want to estimate some function of the average $f(\langle x \rangle)\equiv f(X)$. My first guess was $\...
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1answer
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Finding an unbiased estimator for the variance

While doing some statistics exercises I found a question that I don't know how to solve. The question is as follows: Let $X_{1},...X_{n} \stackrel{iid}{\sim} N(0,\sigma^2)$, where $\sigma^2$ $\in R^{+...
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Testing simple Null vs simple Alternative.

Let $X$ be a random variable with pdf $f$. Suppose we want to test $$H_0:f=f_0\quad $$ against $$ H_1:f=f_1$$. For any $\alpha>0$, define $$T_\alpha(X)= 1\qquad if\quad f_1(X)>c(\alpha)f_0(X)$$ ...
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Estimate parameters of Wishart matrix.

Given a sequence of real Wishart matrices $W_1 , \cdots , W_k \sim \mathcal{W}_m(n,\Sigma)$ where $\Sigma$ is a singular matrix. Are there good estimates for the degrees of freedom? The MLE for $\...
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Estimate degrees of freedom in sample variance.

Given a sequence of independent identically distributed random variables $X_1,\ldots,X_m \sim \chi^2_n / n$ is there literature on estimates for the degrees of freedom $n$? In an attempt to find the ...
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1answer
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How to calculate a Bernoulli Distribution problem

I have my statistics exam quite soon and i came upon this question : At the last referendum, $40\%$ of the Italian population supported the constitutional reform. If a random sample of size $n = ...
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Differentiation of a Complicated Integration

Let, $h$ is differentiable function. $\theta_1\in\mathbb{R}$, $\theta_2>0$ $$F(\theta_1,\theta_2) = \int_{\theta_1-\theta_2}^{\theta_1+\theta_2}\int_{\theta_1-\theta_2}^{y}n(n-1)\frac{(y-x)^{n-2}}{(...
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2answers
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Proof for Simple Linear Regression: What am I doing wrong?

I am trying to prove the well known formula for simple linear regression $$SS_{TOTAL}=SS_{MODEL}+SS_{ERROR}$$ i.e $$\sum_{i=1}^n (y_i - \bar{y})^2 =\sum_{i=1}^n (\hat{y}_i-\bar{y})^2+ \sum_{i=1}^n(\...
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2answers
155 views

Tricky coin probability

Can someone please help, I am getting the wrong answer. Consider four coins labelled as 1, 2, 3 and 4. Suppose that the probability of obtaining a ‘head’ in a single toss of the 𝑖-𝑡h coin is $𝑖/4,\...