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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Guessing Mathematical Probabilities by Tests

I'm stuck with a (maybe simple) problem. I have 4 values possible for a test, and I can do as many tests as I want. What is the minimum number of tests required to be at least at 95% sure I have the ...
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1answer
11 views

Relation between estimator's consistency and biasedness

I have two quick question: If an estimator is consistent, does that imply it is unbiased? If an estimator is biased, does that imply it is not consistent? we know that consistency means ...
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38 views

How to model my not knowledge of distributions. Example form Optional Stopping.

I'm working on a problem related to the secretary problem. Let me give a short overview on the topic I research: You are supposed to choose the best item presented to you in a row of n items. Any ...
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1answer
6 views

Missing approximation to get the Maximum A Posteriori (MAP) estimator of event times with a sparse prior

Assume that a signal $ y $ is a noisy perturbation of time-shifted copies of a given waveform $ f(t) $ defined on K time bins $ \{ 0, \cdots, K-1 \} $: \begin{equation} \forall t \in \{1, \cdots, ...
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1answer
10 views

Maximum a Posteriori (MAP) Estimator of Time Shifts with Poisson Process Prior

Assume that a signal $ y $ is a noisy superposition of time-shifted copies of a given waveform $ f(t) $ on a finite time interval $ [0, T] $: \begin{equation} y(t) = \sum_{i=1}^{n} f(t - \tau_j) + ...
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61 views

According to Chebyshev's rule, how many observations should lie within one and a half standard deviations of the mean?

Using the formula : $p = 1 - k^{-2}$ I calculated that $p = 1 - 1.5^{-2} = 0.56$ , which equals to $56\%$. Because I have $24$ data points I go ahead and solve the number of points is $56\%$ of ...
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16 views

How should I calculate the MLE based on a random sample from $PAR(\theta,2)$

Consider a random sample of size $n$ from a Pareto distribution, $X_i \sim PAR(\theta, \kappa =2)$. I have to compute the MLE, $\hat \theta$, to three decimale places. So I started doing the ...
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Expectation or Integration of the normal cdf

Can any one help me how to solve this pronbelm? I have a random variable $W$, i.e., $$W=\Phi(X)^k\Phi(-X)^m=P(Z\le X)^kP(Z \ge X)^m,$$ $X$ is Normal($\mu$,1), $Z \text{ is Normal(0,1)}$, and $k$ ...
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32 views

Finding independence of two variables

I am trying the following problem: Let $(X_1, Y_1)\ and\ (X_2, Y_2)$ be random points on the plane such that $X_1, X_2, Y_1, and\ Y_2$ are independent $N(µ, σ^2)$. Let $D^2\ $ denote the squared ...
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21 views

sequential anova r

I am a really confused. Assume we have a multiple regression model: $$ y = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} +...+ \beta_{k}x_{k} $$ Using R we can make a test: $$ H0: \beta_{1} = ...
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The conditional probability density function with a specific condition [on hold]

Assume the following discrete time model: $x(t+1)=Ax(t)+w(t)$ where $w(t)$ is zero mean, iid white noise with bounded covariance matrix $Q$. Let $s=x(t)+x(t-1)+x(t-2)$. How I can find ...
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17 views

Finding Asymptotic Confidence Interval with a condition

I am trying to solve the following problem: Let $X_1, X_2$, and $X_3$ be random variables from the following joint pmf: $$f_{X_1,X_2,X_3}(x_1, x_2, x_3) = \frac{n!}{x_1!x_2!x_3!} p_1^{x_1} ...
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913 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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47 views

How to calculate the probability that $X_n$ is not the largest observation in the sample?

I am trying to solve the following problem: Let $X_1,\dots, X_n$, where $n > 4$, be independent random variables such that $X_i ∼ N(i, i)$ for $i = 1, \dots, n$. Let $\bar{X} = ...
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18 views

finding the sufficient and ancillary statistics

I am trying to find the sufficient and Ancillary statistics for the following problem: suppose $(X_1, Y_1). . . ,(X_n, Y_n)$ be iid random vectors from the pdf: $f_{X,Y} (x, y) = {\frac{1}{2π ...
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59 views

Product of two uniform random variables/ expectation of the products

Suppose I want the expectation, $E\Phi(X-\mu)\Phi(\mu-X)$, where $\Phi(.)$ represents the Normal CDF, and X is $Normal(\beta,1)$. Consequently $\Phi(.)$'s are uniform[0,1] and at the same time two ...
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75 views

Estimating the radius of a circle

I have a circle iwth radius $r$. I want to test the hypothesis that $r \leq 2$ vs. $r >2$ based on the posterior of $r$. $r$ follows the prior distribution: $f(r) = \frac{2}{r^{2}}$, $ r >0.5$. ...
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84 views

How to solve an integratation involved an unknown function?

Can anyone have any suggestions how to solve this equation for $w_i$, that is, what is the solution of $w_i$? $$ \int_0^\infty e^{\Phi^{-1}(w_i)ε_i}P(r_i│ε_i )f(ε_i )dε_i=δ $$ Where, $f(ε_i)$ is the ...
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28 views

How to find expectation of Binomial Mass Function?

For example, $$ E \scriptstyle\binom{n}{r}\Phi(X)^r(1-\Phi(X))^{n-r} $$ Where X follows normal distribution with mean $\mu $ and standard deviation 1, and $\Phi(.)$ is the normal CDF. Thank you
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398 views

How do I calculate a point estimate of the largest 10%?

Here is the provided data: The question asked is: Calculate a point estimate of the value that separates the largest 10% of all values in the thickness distribution from the remaining 90% and state ...
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21 views

Risk function for vectors

How do apply risk functions to vectors? Here is the problem I have encountered: Let $X = (X_1, X_2, . . . , X_p)$ be a collection of independent random variables with $X_i \sim N(\mu_i, 1)$ for $i = ...
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25 views

How do i find the sample standard deviation?

Question: The following data were drawn from a normal population:4, 8, 12, 11, 14, 6, 12, 8, 9, 5. estimate the population mean with 90% confidence. i understand how to go about the problem and i ...
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Shows Weibull distribution belongs to a one dimensional exponential family

It is given that $f_\eta(y) = h(y)exp(\eta T(y)-A^*(\eta))$ $P_Y(y)= \frac{k}{\lambda} (\frac{y}{\lambda})^{k-1}exp(-(\frac{y}{\lambda})^k)$ What i did was by arranging $P_Y(y)$ to get ...
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11 views

what's the z parameter in LCA formula?

In this paper on LCA, http://members.home.nl/jeroenvermunt/hagenaars2002b.pdf I can understand the basic eqn. in terms of disease (theta) with symptoms (y). $$ f(\mathbf{y}_i|\theta)= \sum_{k=1}^K ...
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11 views

Conditions for using a t-test for means: contradictory or not?

Besides the conditions of independence and 10%, my teacher says that we need an approximately normal distribution before performing a t-test. My question is: Is this contradictory because the math we ...
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1answer
26 views

Hypothesis testing: normal vs. non-normal

I have the following hypothesis testing problem: $$H_0:X=Y,\quad\text{vs.}\quad H_1:X=Y+Z$$ where $Y\sim\mathcal{N}(0,\sigma^2)$ and $Z$ is a random variable with non-normal continuous ...
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Is there a way to know when a minority of the data is telling the truth?

I am working with temperature data obtained from a network, and i need to identify when temperature sensings are legitimate(Haven't been modified by someone). To find this out, i must compare new ...
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How many people should I ask if a statement (A) is true if the same can be inferred by asking two other statements (X and Y implies A)?

I am asking a number of participants if they believe a given statement is valid. I have a number of such statements, some of which can be inferred. In the made up example below, ...
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1answer
150 views

Probability about confidence interval

Let $X_1,...X_n$ be iid $N(\theta,1)$. A 95% confidence interval for $\theta$ is $\overline{X}\pm\frac{1.96}{\sqrt{n}}$.Let p denote the probability that an additional independent ...
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Necessary to find an estimator's probability distribution before calculating its expectation?

Where $X_{1}, X_{2}, \dots X_{n}$ is an iid distribution with pdf given by: \begin{cases} \frac{1}{\theta}x^{1-\theta} \qquad &\text{If $0 \leq x \leq 1$} \\[5 pt] 0 \qquad &Otherwise ...
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Conceptual/Notational question on conditional distributions and “given”

So in the book I'm reading, I see the notations $f(x|\theta)$ being used to refer to population distributions, dependent on $\theta$ which are in a family. The author explains this as a notational ...
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80 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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The Cramer-Rao Lower Bound proof

Let $X_1, . . . , X_n$ be i.i.d. with density function $f (x|θ)$. Let $T = t (X_1, . . . , X_n)$ be an unbiased estimate of $θ$. Then, under smoothness assumptions on $f (x|θ)$, $$Var(T) >= ...
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Question on sufficient complete statistics proof and estimators of zero

I am trying to prove theorem 7.3.23 in Casella and Burger. Theorem: Let T be a complete sufficient statistic for a parameter $\theta$, and let $\phi(T)$ be any estimator based only on T. Then ...
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evaluation of type1 and 2 error for hypothesis test of variance (two methods with different results)

$\newcommand{\tend}[1]{\oalign{\mbox{\boldmath$#1$}\crcr\hidewidth$\scriptscriptstyle\sim$\hidewidth}}$ Each of $n=10$ persons used the same instrument to measure the same object; the true value ...
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29 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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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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20 views

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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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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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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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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55 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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660 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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28 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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1answer
19 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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30 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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29 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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30 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 ...