Mathematical statistics is the study of statistics from a mathematical standpoint, using probability theory as well as other branches of mathematics such as linear algebra and analysis.

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Finding the liklihood ratio test of that $H_0:\theta =1$ against $H_1: \theta=2$ for two samples with distribution $f(x;\theta)=\theta x^{\theta -1}$

The questions says: A sample of two independent distributions $(x_1,x_2)$ is taken from a distribution with density $f(x;\theta)=\theta x^{\theta -1}$. The test is $H_0:\theta =1$ against $H_1: ...
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1answer
17 views

Explanation of limitations of mean deviation?

My text says ' In a series, where the degree of variability is very high, the median is not a representative central tendency. Thus, the mean deviation about median calculated for such series can not ...
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1answer
18 views

Chi Square “array size”

Hi I have a simple question. I don't understand if a chi square test needs to have an initial square array with the same number of rows and columns : enter image description here Or if the number of ...
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24 views

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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1answer
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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31 views

Asymptotics of a mean of exponential terms involving Gaussians

Let $X\sim \mathcal{N}(0,I_p)$ and $\tau=\sqrt{(2-\varepsilon)\log p}$ and $\varepsilon>0$. I want to prove that for sufficiently small $\varepsilon>0$ the following holds: $$ \mathbb{E}\left[ ...
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26 views

subaddivity of VaR

It is known that the VaR (Value at risk) doesn't fulfill subadditivity, i.e. $VaR(X)+VaR(Y) \le VaR(X+Y)$. But for elliptical distributions subadditivity is true. Questions: (1) Which ...
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17 views

How do you set the half-life of an exponential weight?

I'm trying to follow the process described in a paper, where they smooth a time series using exponential weights, for which I'm using the formula $z_i=\lambda \bar x_i + (1- \lambda) z_{i-1}$, as ...
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1answer
15 views

How is the first term weighted in an exponential smoothing, or exponentially weighted moving average?

I'm working with exponential smoothing for a series, using the formula $z_i=\lambda \bar x_i + (1- \lambda) z_{i-1}$, as explained in this wiki article. Using this method, should $z_{1}$ be defined ...
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29 views

Calculate variance and expected value

I am trying to do some combinations when calculating variance and expected value, but i am stuck, i dont know how to combine them. the two functions look like this: i calculate the variance for ...
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15 views

Using optim and fitdistr in R to find parameters

I am using R to fit distributions. I have been given the data and have been asked to find the optimised parameters(for lognormal, weibull, exponential and gamma functions) using: 1) fitdistr and 2) ...
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22 views

How much can we trust the estimation (probably simple)

To begin with, I don't know whether this task is correct. Suppose we have a conveyor with bombs and each bomb is examined by $n$ minions. Minion can either accept it, reject it or stay uncertain. ...
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46 views

How to use sub-pattern frequencies to calculate the authenticity the main pattern

I'm trying to determine the probability that something is real by comparing the frequency of the sub-permutations it contains. Here is an example: You want to guess whether or not a random symbol is ...
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19 views

Need help with differentiating between constant comparative method and deviant case analysis..

Questions is as follows: On page 341 the researchers stated, “As hypotheses develop regarding the relationships between categories, it is necessary to determine whether the data support the ...
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1answer
19 views

Mean and variance of the sum of cgf gamma and poisson distribution

Suppose that we have the sum of two cumulant generating function: $\log e^{m(e^t-1)} + \log(1-dt)^{-c}$, and we wish to find the expectation and variance without differentiation. I realize ...
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11 views

Quasi-Monte Carlo with Conditional Distributions

I want to estimate $E(f(X))$ using quasi-Monte Carlo where $X = (X_1,\ldots,X_n)$ is a random vector and $$ X_i\sim f(\cdot; \theta), \quad \text{independent}, $$ where $\theta \in \mathbb{R}$ is some ...
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0answers
12 views

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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14 views

expected 1 norm of a normal vector

Given a normal vector $X$ such that $\mathbb{E}(X)=0$ and $Cov(X)=Id$, is it possible to get an expression for $$\mathbb{E}(\|A X\|_1)$$ where $A$ is a given matrix. I know that in dimension 1, we ...
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33 views

Calculate number of trials reaching $p_k$ probability for $k$ successes given the $p_t$ probability of each trial success

Basically, I'd like to be able to answer questions in the form of "What is the number of trials needed to have at least $p_k$ probability of at least $k$ successes, given that on each trial the ...
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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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1answer
31 views

Understanding the value of standard deviation

I have two datasets $\{10,10,2,2\}$ and $\{13,7,0,4\}$. Now, when I compute standard deviation for both the sets, I get $4$ and $4.74$ respectively. My question: what is the significance of $4.74$ or ...
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23 views

Basic Asymptotic Theory book

I would love if someone can recommend me a book where Basic Asymptotic Theory is thoroughly covered and explained with some examples. I'm currently reading Econometric Analysis of Cross Section and ...
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1answer
26 views

joint pdf of $X$ and $Y$ find constant

$$f(x, y) = ce^{−x−y}$$ for $0 \le x \le y < \infty$, Calculate the value for $c$ that makes $f$ a valid pdf. how do you find $c$ when the domain of $x$ and $y$ both contain each other?
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14 views

An example where pearson is wildy different to spearman? [duplicate]

Im looking to spearman and pearson, and from what i understand spearman is better at looking at curves. Can i see an example of a small set of data (10 or less) where this difference is large.
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5 views

When to different hypothesis testing concepts

For hypothesis testing Am I right in saying that I use the Z test , if I know the two populations are independent and the two population’s standard deviations are known the pooled t-test, if the ...
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1answer
32 views

Variance of a random variable [closed]

How do you get the variance of a random variable $X$ where $X = \frac{1}{6}(A \cdot B)$ and where $A$ and $B$ are two independent random variables with variances $\sigma_A^2$ and $\sigma_B^2$, ...
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1answer
22 views

Find the cdf $F_{X,Y}(u,v)$ if the pdf is given by $f_{X,Y}(x,y) = 6x$ for $0\leq x \leq 1$ and $0 \leq y \leq 1-x$

Find the cdf $F_{X,Y}(u,v)$ if the pdf is given by $$f_{X,Y}(x,y) = 6x$$ for $0\leq x \leq 1$ and $0 \leq y \leq 1-x$ I have the solution to this, but I don't understand it completely. Can some one ...
2
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2answers
21 views

Find $P(Y=5)$ for $Y=X_1+X_2+X_3$, where $X_i$ are mutually indpt Poisson R.V

In this problem, I am told that $X_1,X_2,X_3$ are mutually independent Poisson random variables with means $2,1,4$ respectively. I am also told to find the moment generating function for ...
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2answers
93 views

Suppose that $E[X^n] = 3n$. Find $E[e^X]$…

Suppose that $E[X^n] = 3n$. Find $E[e^X]$. Hint from my professor: $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} +···$ Not quite sure how to solve this problem, wouldn't $e^x$ go on exponentially. ...
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1answer
24 views

Use cdf to find expectation

I have a cdf for a $\mathbf {discrete}$ random variable, $X$, $$F_X(x)=1-(1-p)^{xn}$$ where $p\in(0,1)$, $n\in\mathbb N$, $x\in\mathbb N$ My thought is to use $$E[X]=\sum_{x=0}^\infty ...
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1answer
37 views

Let X be any random variable. Find $\displaystyle\lim_{b\to-\infty} P[X \le b]$…

Let $X$ be any random variable. Find $\displaystyle\lim_{b\to-\infty} P[X \le b]$ I would think $b$ is zero, making this an infinite sum but really not sure. Any help/direction with this problem is ...
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21 views

Show that if X is a continuous r.v. and it takes only positive values then $E(X) =\int_{0}^{∞}Z P[X ≥ t] dt $ [duplicate]

Show that if X is a continuous r.v. and it takes only positive values then: $$E(X) =\int_{0}^{∞} P[X ≥ t] dt$$ I am not really sure how to begin this proof. Any help or insight would be ...
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1answer
23 views

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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12 views

Find which categorical parameters have correlation with a given property [migrated]

I am a molecular biologist, so I apologize in advance, if the question is too basic. Example described below is for simplicity (I have a similar situation with my data to the one I describe below) I ...
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8 views

If a family of densities is not complete then is it necessary that there isn't any MVUE?

The question is about the truth of this statement: "If the family $\{f(x;\theta):\theta\in\Omega\}$ is not complete, then there doesn't exist any MVUE" MVUE is an abbreviation for "Minimum Variance ...
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9 views

Test the hypothesis that passing/failing this module is independent of the number of attendances

Marks ≥ 40% result in a pass, while marks < 40% result in a fail. Test the hypothesis that passing/failing this module is independent of the number of attendances The table for this question has ...
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0answers
13 views

Anti-intuition when finding statistical model of a random variable using Maximum Entropy Principal

I was trying to understand the Maximum Entropy Principal, and was calculating a very simple example, but ran into some confusion. Consider a random variable $X$, which can only take values $1,2$ and ...
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24 views

Consequences of fitting a regression model with an intercept term when it should be through the origin

Suppose a true model is $Y_i=\beta X_i +e_i$, where $e$ is the random error. Suppose instead we fit the model (using least squares) as $Y_i=\alpha_0+\alpha_1 X_i +v_i$, where $v$ is the random error. ...
3
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1answer
40 views

98% confidence interval

For this question from a large amount of data I have calculated that the mean is 44.22, the sample size is 100 and the standard deviation is 22.0773. From this I am asked to , make the 98% confidence ...
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23 views

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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7 views

Without homoscedasticity, is OLS still the best estimator (aka BestLinearUnbiasedEstimator…BLUE)?

Consider the Gauss Markov assumptions. Suppose we have a random sample $\lbrace x_n,y_n \rbrace_{n=1}^{N}$. Assume for a simple linear regression model $y_n = \beta_0 + \beta_1 x_n + \varepsilon_n$ we ...
2
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1answer
29 views

Compute the cumulative distribution function of the variable $R=\sqrt{X^2+Y^2}$

I've returned to the study of statistics after a long while and I'm trying to solve some problems. One of those is the next: Suppose $X$ and $Y$ are random independent variables with normal ...
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1answer
61 views

Forward price in Black Scholes Model

Recall that a forward contract on $S_T$ contracted at time $t$, with time of delivery $T$, and with forward price $f(t; T, S_T)$ can be seen as a contingent T-claim $X$ with payoff: $$ X = S_T - f(t; ...
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1answer
19 views

Hypothesis testabout the variability of 2 Samples

The hydrocarbon emissions are known to have decreased dramatically during the 1980s. A study was conducted to compare the hydrocarbon emissions at idling speed, in parts per million (ppm), for ...
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1answer
26 views

What is the difference in how $\mathrm{R}^2$ and $\mathrm{R}$ values are interpreted?

In statistics, there is the $\mathrm{R}$ value for the product moment correlation coefficient and the $\mathrm{R}^2$ value for the coefficient of determination. In both cases they are described as a ...
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How is the $R^2$ value for exponential regression calculated if not by product moment correlation coefficient?

I am analysing some $x$ and $y$ values using Excel by plotting them on a graph and adding a line of best fit then using the equation for the lines of best fit. The exponential line of best fit has a ...
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11 views

Choosing between two exponential hypothesis

Problem. Let $P$ have density $e^{−x}$ on $[0,\infty)$ (so $P$ is a standard exponential distribution) and let $Q$ be the distribution of $X + 1$ where $X$ has distribution $P$. What is the maximum ...
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1answer
26 views

Hypogeometric Probabilities not adding to one

When I was computing the Hypogeometric formula probabilities I was having a little difficulty following the solutions manual. I attempted to compute the event where there was one Red Chip and One ...
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8 views

Why does the difference between $\mathcal{\hat{b}}$ and $\mathcal{b}$ converge to a normal distribution in a SLRM?

We are considering simple linear regression models in my econometrics class. We are given a random sample $\lbrace x_i,y_i \rbrace_{n=1}^\infty$ $$Y_i = b_0 + b_1 X_i + \varepsilon_i$$ where ...
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3answers
44 views

How does using one distribution as another's sample size affect variance?

How does using one distribution as another's sample size affect variance? For example, let's say I roll a 6-sided dice and record the number shown. Then, I roll 'that many' 6 sided dice more and ...